Republic of Iraq Ministry Of Higher Education & Scientific Research University Of Baghdad College Of Engineering Civil Engineering Department

A THESIS
SUBMITTED TO THE CIVIL ENGINEERING DEPARTMENT OF THE COLLEGE OF ENGINEERING OF THE UNIVERSITY OF BAGHDAD IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN CIVIL ENGINEERING ( STRUCTURES )

Approved by the Dean of the College of Engineering.
Signature : Name : Prof. Dr. Ahmed Abdul Saheb Mohammed Ali Dean of The College of Engineering University of Baghdad Date : 29 / 9 / 2014

ACKNOWLEDGMENTS

All

the praises and thanks be to Allah, Lord of the world,

The Beneficent, the Merciful, for His grace and mercy. Who hath guided us to this, we could not truly have been led aright if Allah had not guided us.

I would like to express my gratitude to my supervisor Prof. Dr. Adnan
Falih Ali for the useful comments, remarks and engagement throughout the preparation of this M.Sc. thesis. Furthermore I would like to thank Ms. Shatha Dheia for introducing me to the topic as well for the support on the way.

Also, I would like to thank the participants in my survey, who have
willingly shared their precious time during the process of interviewing. And I would like to thank my loved ones, who have supported me throughout the entire process, both by keeping stead fast and helping me in putting pieces together. I will be grateful forever for their love.

Finally, my special thanks are expressed to my father, mother,
dear wife, brothers, sisters and my best friends for their endless support and encouragement at all times. O Allah, bless and peace upon Prophet Muhammad and his progeny; and the conclusion of my prayer will be: Praise be to Allah, Lord of the world!

feras temimi 2013

I

ABSTRACT The
study presents a simplified procedure for developing an idealization

of curved cellular type bridge decks under free vibration and dynamic loads caused by earthquake based excitation. Two cases of curved cellular type bridge decks are analyzed single and multiple cells. The procedure is applicable to rectangular and trapezoidal cellular sections of equal and unequal dimensions of cells. This proposed idealization procedure is designated as the Panel Element Method (PEM). The proposed idealization technique is based on a modified element called "Panel Element (PE)", which is capable of representing in-plane and out-of-plane shear and flexural stiffness, in two orthogonal directions, axial stiffness in addition to the coupled torsional warping stiffness through few local and global degrees-of- freedom that assess the aforementioned global behavior. In the proposed Panel Element (PE), the curved cellular bridge deck is idealized as an assemblage of solid planar and non-planar plate units that are developed on the basis of the three dimensional wide column analogy which laces in consideration of the effect of kinetic and strain energy (both in-plane and out-of-plane) and the effect of shear deformation (in-plane and out-of-plane). Each plate unit or panel element is assumed to be composed of two rigid arms centrally connected by a flexible member with four nodes located at the boundaries of the rigid arms. The proposed element "Panel Element (PE)", is coded as Component Element (CE) and has proved to be capable of modeling a full plane panel of a curved cellular deck in its three-dimensional behavior by one element only. For verification purpose and to demonstrate the range of applicability of the new idealization technique, a comparative study was made with the Finite Element Method (FEM), as a standard procedure, used to idealize the cellular bridge decks.
II

Different configurations of curved cellular bridge decks are considered to provide a thorough understanding of the dynamic behavior of the curved bridge deck when acted upon by earthquake based excitation besides the validation purposes. A computer program using (MATLAB R2012b) is specially written using the proposed algorithm of the new idealization technique to solve the Eigen-value problem of natural frequencies and mode shapes in vertical, torsional and lateral movements, also earthquake analysis results. Comparison was made with those evaluated by the finite element approach using the ready software (ANSYS 12.0) to check the adequacy and suitability of the proposed element in analyzing the cellular concrete bridge decks. The results show the proposed Panel Element Method (PEM) can predict accurately the free vibration characteristics (natural frequencies and mode shapes) of single and double cell curved box-girder bridge decks, and for rectangular and trapezoidal cross-sections. The difference in the fundamental modes of vibration (natural frequency) is less than (7%) between the proposed panel element (PE) and the standard finite element (FE). Also, the Panel Element Method (PEM) has proved to be valid in estimating the earthquake response for both cases of single and double cell bridge decks, for all the ranges of the aspect ratios; the results obtained by the Panel Element Method (PEM) are acceptable, with an error of less than (12%) in deflection and less than (18%) in moments and shear forces for the cases of very large aspect ratios. The study demonstrates the validity of the proposed method "Panel Element Method (PEM)" with wide range of applicability for the dynamic behaviors of free and forced vibration response analysis and the approximate earthquake response analysis of the curved cellular type bridge decks of different configurations.

CHAPTER SIX (6.1) Maximum Response of a Cantilever Bridge Deck of a Single and Double Cells (Base Excitation in X-Direction) ……...………….... (6.2) Maximum Response of a Cantilever Bridge Deck of a Single and Double Cells (Base Excitation in Y-Direction) ……...……….... (6.3) Maximum Response Variation with (tw/ts) Ratio for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) ………………...…………………………………. (6.4) Maximum Response Variation with (tw/ts) Ratio for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) ………………...…………………………………. (6.5) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in X-Direction) ……………...…..……………… (6.6) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) ……………...…..……………… (6.7) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in Y-Direction) ……………...…..……………… (6.8) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) ……………...…..……………… (6.9) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in X-Direction) ………….…………………………… (6.10) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) ………….……………….……………………… (6.11) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in Y-Direction) ………….…………………………… (6.12) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) ………….……………….………………………

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XII

NOTATIONS
The major symbols that are used in the text are listed below, the other notations (not listed here) are defined, as they first appear, otherwise, the notation used is clearly defined within the text. General Symbols [A]T , {a}T [A]-1 ,d | | , det. { } [ ] Transpose of matrix [a] and vector {a} Inverse of matrix [a] Partial and full differentiation symbol Determinant of matrix or absolute value Vector or column matrix Matrix Scalar [0] [1] a A Ae b ! C [C] d dt D E {e} Fs Fsmax Fs1max {Fsmax} {FSnr} {F(t)} G h Zero Matrix One Matrix Acceleration Cross-sectional area of the panel element that normal to direction Effective shear area of the panel element cross-section The width of the cross-section of the deck A constant depends on (D/t) ratio Cos Damping matrix of the system The height of the cross-section of the deck The tolerance number Width of the Panel unit element Modulus of elasticity (Young's Modulus) of the panel unit material Displacement vector in the static approach The elastic force The maximum force The maximum response The maximum force vector Equivalent external force associated with peak displacement in r-direction at mode (n) The applied load vector Shear modulus of the panel unit material Length of the panel element along the longitudinal axis of the deck
XIII

Moment of inertia of panel cross-section about the local x-axis Moment of inertia of panel cross-section about the local y-axis Torsional moment constant of panel element about the local z-axis The torsional moment of inertia of the panel element The polar moment of inertia of panel element about the local z-axis K parameters of stiffness matrix of the panel element Stiffness coefficient of corresponding to (u) d.o.f Stiffness coefficient of corresponding to (v) d.o.f Stiffness coefficient of corresponding to (w) d.o.f Stiffness coefficient of corresponding to ( L) d.o.f Stiffness coefficient of corresponding to ( Z) d.o.f Stiffness matrix of the whole system Stiffness matrix of the standard space frame element Global stiffness matrix of the component element Stiffness matrix of the panel element in local coordinates Stiffness matrix of panel element in global coordinates Stiffness sub-matrix of panel element corresponding to d.o.f at node (i) Stiffness sub-matrix of panel element corresponding to forces at node (i) due to displacements at node (j) The transpose matrix of [kij] matrix Stiffness sub-matrix of panel element corresponding to d.o.f at node (j) Stiffness matrix of the individual component element in local coordinates Span of the bridge deck Modal earthquake excitation factor in r-direction at mode (n) The projection of length onto X-Y plane mass of the structure Moment Total mass of the whole bridge deck The diagonal coefficients of the mass matrix Generalized mass at node (n) The total mass of the panel element The number of longitudinal displacement at each diaphragm position Mass coefficient of corresponding to (u) d.o.f Mass coefficient of corresponding to (v) d.o.f Mass coefficient of corresponding to (w) d.o.f Mass coefficient of corresponding to ( L) d.o.f Mass coefficient of corresponding to ( Z) d.o.f Mass matrix of the whole system The consistent mass matrix of the standard space frame element Global mass matrix of the component element Mass matrix of the panel element in local coordinates Mass matrix of panel element in global coordinates
XIV

Mass sub-matrix of panel element corresponding to d.o.f at node (i) Mass sub-matrix of panel element corresponding to inertia at node (i) due to accelerations at node (j) The transpose matrix of [Mij] matrix Mass sub-matrix of panel element corresponding to d.o.f at node (j) The lumping mass matrix of the standard space frame element Mass matrix of the panel element within effect of diaphragms in local coordinates Mode number The number of degree of freedom d.o.f per one section Number of elements Number of panels Number of cells Shear factor of the panel element cross-section in local x-direction Shear factor of the panel element cross-section in local y-direction Number of all panel elements An earthquake influence vector Radius of gyration of the panel cross-section about the local z-axis Sin Spectral acceleration (Pseudo-acceleration) Pseudo acceleration in r-direction A specific number The thickness of the panel element Natural Period Distance between master node and the top node for both the in-plane and out-of-plane elements The 3d-coordinates rotation sub-matrix of the panel element Transformation matrix to the global coordinate of the bridge Transformation matrix of the panel element The transpose matrix of [Te] matrix Transformation sub-matrix of the panel element Transformation matrix of constrained within diaphragms in global coordinates The transpose matrix of [Tg] matrix Transformation sub-matrix of the panel element within diaphragms The 3d-coordinate transformation matrix of the panel element The transpose matrix of [TR] matrix The slab thickness of the cross-section of the deck The web thickness of the cross-section of the deck Translation displacement of the cellular bridge system (in vertical plan perpendicular to the longitudinal axis of the decks) Initial trial vector The time varying base acceleration component in the r-direction Displacement vector of the standard space frame element Displacement vector of the panel element
XV

Vector of displacement in element Cartesian coordinates Vector of displacement in global Cartesian coordinates Time depended displacement vector Mode shape of the system Time depended acceleration vector Time depended velocity vector Peak values of displacement in r-direction Shear force The earthquake-response integral Translation displacement of the cellular bridge system in Z-direction Local Cartesian coordinates system Local Cartesian coordinates system Global Cartesian coordinates system The x, y and z coordinates of the node no. (1) of the panel element The x, y and z coordinates of the node no. (2) of the panel element The horizontal dimension between the master node and the far left node for both in-plane and out-of-plane elements The horizontal dimension between the master node and the near right node for both in-plane and out-of-plane elements The vertical dimension between the master node and the bottom node for both in-plane and out-of-plane elements Generalized coordinate amplitude The vertical dimension between the master node and the top node for both in-plane and out-of-plane elements Weight density of panel element material (kN/m3) Damping ratio Angle between a horizontal line at the cellular bridge cross-section and the direction of the panel element The phase angle Local rotational displacement of deck system about the minor axis of each individual panel element The user-selected adjustment angle of the rotation of each panel element in direction of longitudinal of the bridge Rotational displacement of deck system about the longitudinal Z-axis The vector of Eigen values Constant parameter between 0 and 1 Mass density of panel unit material (kg/m3) Shear stress Poisson's ratio of panel element material Natural frequency of the cellular bridge Natural frequency at mode (n) Mode shape vectors of deck system Mode shape vectors at mode (n)

The continuous expansion of highway networks throughout the world is largely
the result of great increase in traffic, population and extensive growth of metropolitan urban areas. This expansion has led to many changes in the use and development of various kinds of bridges. The bridge type is related to providing maximum efficiency of use of material and construction technique, for particular span, and applications. As the span increases, dead load is an important increasing factor. To reduce the dead load, unnecessary material, which is not utilized to its full capacity, is removed out of section, this results in the shape of box girder or cellular structures, depending upon whether the shear deformations can be neglected or not. Span range is more for box bridge girder as compare to T-beam Girder Bridge resulting in comparatively lesser number of piers for the same valley width and hence results in economy. A box girder bridge is a bridge in which the main beams comprise girders in the shape of a hollow box, and it is formed when two web plates are joined by a common flange at both the top and the bottom. The closed cell which is formed has a much greater torsional stiffness and strength than an open section and this feature is the usual reason in choosing a box girder configuration. Box girders are rarely used in buildings (box columns are sometimes used but these are axially loaded rather than loaded in bending). They may be used in special circumstances, such as when loads are carried eccentrically to the beam axis "When tension flanges of longitudinal girders are connected together, the resulting structure is called a box girder bridge". Box girders can be universally applied from the point of view of load carrying, to their indifference as to whether the bending moments are positive or negative and to their torsional stiffness; from the economy point of view.
1

Chapter One

Introduction

The box girder normally comprises either pre-stressed concrete, structural steel, or a composite of steel and reinforced concrete. The box is typically rectangular or trapezoidal in cross-section. Box girder bridges are commonly used for highway flyovers and for modern elevated structures of light rail transport. Although normally the box girder bridge is a form of beam bridge, box girders may also be used on cable-stayed bridges and other forms. And if made of concrete, the construction of box girder bridges may be cast in place using false-work supports, removed after completion, or in sections if a segmental bridge. Box girders may also be prefabricated in a fabrication yard, then transported and emplaced using cranes.

1.2 Structural Forms of Bridge Decks
This section reviews and categorizes the principal types of bridge decks that are currently being used. The types of bridge deck are divided into beam, grid, slab, beam-and-slab and cellular, to differentiate their individual geometric and behavioral characteristics. Inevitably many decks fall into more than one category, but they can usually be analyzed by using a judicious combination of the methods applicable to the different types, and the reinforced concrete is particularly well suited for use in bridges of all types due to its durability, rigidity, and economy. The bridge decks can be categorized as follows [58]:

1.2.1 Beam Decks
A bridge deck can be considered to behave as a beam when its length exceeds its width by such an amount that when loads cause it to bend and twist along its length, its cross-sections displace bodily and do not change shape, as shown in Fig. (1.1).

The most common beam decks are footbridges, either of steel, reinforced concrete or pre-stressed concrete. They are often continuous over two or more spans. Some of the largest box girder decks may be analyzed as beam to determine the longitudinal variation of the internal generalized forces, as shown in Fig. (1.2).

1.2.2 Grid Decks
The primary structural member of a grid deck is a grid of two or more longitudinal beams with

transverse beams (or diaphragms) supporting the running slab. Loads are distributed between the main longitudinal beams by the bending
Figure (1.3): Load distribution in grid deck and twisting of the transverse beams, by bending and torsion of beam members. [58]

as shown in Fig. (1.3). Because of the amount of workmanship needed to fabricate or shutter the transverse beams, this method of construction is becoming less popular and is being replaced by slab and beam-and-slab decks with no transverse diaphragms.

1.2.3 Slab Decks
A slab deck behaves like a flat plate which is structurally continuous for the transfer of moments and torsions in all directions within the plane of the plate. When a load is placed on part of a slab, the slab deflects locally in a 'dish' causing a two-dimensional system of moments and torsions which transfer and share the load
3

Chapter One

Introduction

to neighboring parts of the deck which are less severely loaded, as shown in Fig. (1.4). A slab is "isotropic" accepted as sufficiently accurate for most designs.

Figure (1.4): Load distribution in slab deck by bending and torsion in two directions. [58]

One type of deck which does not fit neatly into any of the main categories is the "shear-key" deck. A shear-key deck is constructed of contiguous pre-stressed or reinforced concrete beams of rectangular or box sections, connected along their length by in situ concrete joints, as shown in Fig. (1.5).

1.2.4 Beam-and-Slab Decks
A beam-and-stab deck consists of a number of longitudinal beams connected across their tops by a thin continuous structural slab, as shown in Fig. (1.6a). In transfer of the load longitudinally to the supports, the slab acts in concert with the beams at their top flanges. At the same time the greater deflection of the most heavily loaded beams bends the slab transversely so that it transfers and shares out the load to the neighboring beams, as shown in Fig. (1.6b). Sometimes this transverse distribution of load is assisted by a number of transverse diaphragms at points along the span, so that deck behavior is more similar to that of a grid deck.
4

It is important to mention here the composite concrete slab-steel beam decks, which are also in use. This type is usually constructed using steel I-beams spaced at a certain required distance apart, then a reinforced concrete slab will be cast in situ on their top, as shown in Fig. (1.7).

Figure (1.7): Composite Beam-Slab decks. [40]

1.2.5 Cellular Decks
The cross-section of a cellular or box deck is made up of a number of thin slabs and thin or thick webs which totally enclose a number of cells. These complicated structural forms are increasingly used in preference to beam-and-slab decks for spans in excess of 30m (100ft) because in addition to the low material content, low weight and high longitudinal bending stiffness, they have high torsional stiffness which gives them better stability and load distribution characteristics. To describe the behavior of cellular decks it is convenient to divide them into multi-cellular slabs and box-girders. Multi-cellular slabs are wide shallow decks with numerous large cells. Box-girder decks have a cross-section composed of one or a few large cells, the edge cells often have triangular cross-section with inclined outside web. Frequently the top slab is much wider than the box, with the edges cantilevering out transversely. Excessive twisting of the deck under eccentric loads on the cantilevers is resisted by the high torsional stiffness of the structure.

5

Chapter One The cellular bridge decks can also be classified as follows [41]: a- Shallow Cellular Structure:

Introduction

This type of cellular bridge decks has more than two or three cells, shallow depth, and thin walls as compared with the span of the bridge deck, as shown in Fig. (1.8).

Figure (1.8): Shallow Multi-Cell decks types. [41]

b- Deep Cellular Structure: This type of bridges is frequently used in large structures and presents a large problem. This kind is also known as spin beam bridge. Its large depth reaches (2m) but is still considered as a thin walled as compared to the span of the bridge deck. Fig. (1.9) shows the above deep cellular deck bridges. The typical cross-section of cellular type bridge deck is shown in Fig. (1.10) [127], and typical curvature box girder in Fig. (1.11). Only this type of bridge is considered in this study.

1.3 Static and Dynamic Loads
There are two types of forces/loads that may act on structures, namely static and dynamic forces.

1.3.1 Static Loads on Bridges
Static forces are those that are gradually applied and remain in place for longer time. These forces are either not dependent on time or have less dependence on time. All load systems are defined in terms of one more of three systems of loading, as follows [40]: (A) Lane loads Almost all national codes specify some form of lane loading. This is intended to correspond to the order of value of load to be expected from normal highway traffic. Typical features of lane loadings are: (1) An equivalent uniformly distributed load which may be independent of the loaded length of lane or dependent upon this length (depending on Codes). (2) A concentrated or line load which is positioned appropriately to yield maximum bending moments or shearing forces. (3) Loading from a vehicle or vehicles, transmitted by wheel. (B) Individual abnormal vehicle loads Nearly all countries specify a form of concentrated load for the computation of local stresses. Fewer countries require allowance for the passage of individual heavy vehicles as an abnormal event. (C) Individual wheel or axle loads Supplementary loads are also commonly specified for the determination of local stresses that might occur in bridges. The design of live load for a major bridge is compared in this section with "British Ministry of Transport loading" [20], "American Association of State Highway and Transportation Officials (AASHTO) loading" [12], and "Iraqi standard specification" [72].
8

Chapter One

Introduction

1.3.2 Dynamic Loads on Bridges
A dynamic load is the forces that move or change when acting on a structure, and the dynamic forces are those that are very much time dependent and these either act for small interval of time or quickly change in magnitude or direction. Earthquake forces, wind forces, machinery vibrations and blast loadings are examples of dynamic forces. Structural response is the deformation behavior of a structure associated with a particular loading. Similarly, dynamic response is the deformation pattern related with the application of dynamic forces. In case of dynamic load, response of the structure is also time-dependent and hence varies with time. Dynamic response is usually measured in terms of deformations (displacements or rotations), velocity and acceleration. Dynamic force, F(t), is defined as a force that changes in magnitude, direction or sense in much lesser time interval or it has continuous variation with time. The variation of a dynamic force with time is called history of loading.

1.4 Earthquake Effects on Bridges
Earthquake forces are forces that depend on the geographical location of the bridge. These forces are temporary, and act for a short duration of time. The application of these forces to a bridge is usually studied with their effect upon piers, pile caps, and abutments. There are four factors that should be taken into consideration to determine the magnitude of the seismic forces: 1) The dead weight of the entire bridge. 2) The ground acceleration in three Cartesian directions. 3) The natural frequencies of the bridge structural system. 4) The type of soil or rock layers serving as bearing for the bridge. The sum of these factors is reduced to an equivalent static force, which is applied to the structure in order to calculate the forces and the displacements of each bridge element.
9

Chapter One

Introduction

1.5 Diaphragms in Bridges
A support diaphragm is a member that resists lateral forces and transfers loads to support. Some of the diaphragms are post-tensioned and some contain normal reinforcement. It is needed for lateral stability during erection and for resisting and transferring earthquake loads. Based on previous researches, diaphragms are ineffective in controlling deflections and reducing member stresses. Moreover, it is commonly accepted that diaphragms aided in the overall distribution of live loads in bridges. The main function of diaphragms is to provide stiffening effect to deck slab in case of bridge webs are not supported directly by the top of bearings. Therefore, diaphragms may not be necessary in cases where bridge bearings are placed directly under the webs because loads in bridge decks can be directly transferred to the bearings. On the other hand, diaphragms also help to improve the load-sharing characteristics of bridges. In fact, diaphragms also contribute to the provision of torsional restraint to the bridge deck, as shown in Fig. (1.12) below.

Figure (1.12): Diaphragms in bridges. [121]

10

Chapter One

Introduction

1.6 Objective and Scope
The primary objective of the present work is to develop a new simplified procedure and an alternative reliable idealization technique for dynamic and earthquake response analysis of curved cellular type bridge decks. This new idealization technique should have the following features: 1. The number of unknown degrees of freedom (d.o.f) required to represent the dynamic behavior of any deck plate should be smaller than those required by other methods such as Finite Element method (FEM); and the formulation of the governing equations and their solution may be easily implemented on a computer. 2. The response of individual element of a deck should be directly attainable without the necessity of a secondary analysis. 3. The range of applicability of the proposed idealization method is compared with the free vibration and the earthquake response analysis of certain standard deck configurations, as obtained using the finite element (FE) idealization approach. Different configurations of cellular bridge decks will be studied to evaluate the dynamic behavior and the maximum response.

11

Chapter One

Introduction

1.7 Layout of Thesis
This thesis consists of seven chapters, and the following is a brief outline of the major topics covered in this thesis: In Chapter One, a general introduction and classifications of bridge types are summarized. An exposition of the problem is presented. The objective and scope of the present research work are also included. Chapter Two, contains a literature review that describes the summary of previous research that deals with the dynamic analysis of curved cellular type bridge decks, and previous studies related to the present work. Chapter Three, includes the finite element modeling, and description of the types of elements that are used in this modeling, and the important details that are concerned with idealization of the structure by the Finite Element Method (FEM). Chapter Four, includes the description of the proposed Panel Element (PE) method, which includes the element type, assumptions and derivations of the property stiffness and mass matrices for the element type used to idealize the structure. It also includes a description of the effect of the rigid diaphragms. Also it contains a list of the governing equations of motion that govern the structure behavior and the method of solution implemented in the study. Chapter Five, includes the solution procedure adopted for the undamped free vibration analysis. Numerical examples are presented in this chapter of the validation of the proposed Panel Element (PE) idealization method vis-à-vis the Finite Element (FE) idealization method. Other numerical examples of several parametric studies to develop the range of applicability and some other aspects are also presented in this chapter. Chapter Six, covers the details about the analysis procedure that is used in determining the cellular bridge deck response when subjected to an earthquake base excitation, and an earthquake response analysis of the structure configurations is studied in this chapter. Chapter Seven, includes the major conclusions that are drawn from the present study together with the recommendations for future works on this subject.
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Chapter Two

Literature Review

CHAPTER TWO

LITERATURE REVIEW
2.1 Introduction

In

this chapter a brief review of literature on the most common methods

that contribute to the field of analysis of the curved box girder bridges. It will be presented with special emphasis on analysis of curved box-girder bridges and on the development of the finite element method of analysis. Also it summarizes the static and dynamic behavior of curved box girder bridges and covers a wide range of topics that can be itemized as follows: 1. Literature pertaining to the elastic analysis methods. 2. Methods of idealization used in this work. 3. Behavior of curved box girder bridges. 4. Free Vibration Analysis of bridges. However, literature review for more specific areas will be included in the following sections.

2.2 Review of the Analysis of Earliest Bridges
The development of the curved beam theory by Saint-Venant, (1843) and later the thin-walled beam theory by Vlasov, (1965) marked the birth of all research efforts published to date on the analysis and design of straight and curved box-girder bridges. Many technical papers, reports, and books have been published on various applications of, and even modifications to, the two theories. Recent literature on straight and curved box girder bridges has dealt with analytical formulations for better understanding of their complex behavior. Few authors have undertaken experimental studies to investigate the accuracy of the existing methods.
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Chapter Two

Literature Review

Box-girder bridges are analyzed using various analytical and numerical methods. Simple solution for curved box-girder bridges is only available in special cases. The curvilinear nature of this type of structures along with the complex deformation pattern and stress fields developed under full and partial truck loading conditions leads the designer to adopt methods of analysis limited in accuracy and scope of application due to their inherent simplifying assumptions [11].

2.3 Analytical Methods for Box Girder Bridges
There are several methods available for the analysis of box girder bridges. Besides, AASHTO specifications also recommended analyzing box-girder bridges. In each analysis method, the three-dimensional bridge structure is usually simplified by means of assumptions in the geometry, materials and the relationship between its components. The accuracy of the structural analysis is dependent upon the choice of a particular method and its assumptions. The highlights of the references pertaining to elastic analysis methods of straight and curved box girder bridges are published by Sennah and Kennedy, (2002) [114]. A review of different analytical methods for box girder bridges has been presented by Samaan, (2004) [107]. Aldoori, (2004) [6], has discussed the theoretical aspects of some of the methods. A brief review of the analytical methods of box girder bridges is presented below.

2.3.1 Orthotropic Plate Theory Method
This method is a two dimensional method. An orthogonal-anisotropic plate simulates the deck with different specified elastic properties in two perpendicular directions. The orthotropic plate theory method considers the interaction between the concrete deck and the curved girder of a box girder bridge. In this method the stiffness of the diaphragms is distributed over the girder length and the stiffness of the flanges and girders is lumped into an orthotropic plate of equivalent stiffness. The orthotropic plate theory for the analysis of simply supported cellular bridge decks with rectangular plane geometry was first used by Little, (1954) [83], who analyzed simply supported cellular bridge decks with rectangular plane
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Chapter Two

Literature Review

geometry, which was developed by considering a new torsional constant of the section by Little and Rowe, (1955) [82], and by Cusens and Pama, (1969) [40], who have further developed Little's work by considering a new method in calculating the torsional constant of the section. Massonnet and Gandolfi (1967) [89], propose a simplified solution for rectangular orthotropic plates supported on opposite edges. Robertson, Pama and Cusens, (1970) [104], propose a relatively simple solution for the analysis of cellular bridge decks as rectangular shear-weak plates (The steel plates can be designed as a thin weak plate in shear). Mohammed, (1994) [91], idealized the cellular plate structure as an equivalent orthotropic plate. A mathematical model is derived to represent the orthotropic plate element by a grid of beams with specified elastic rigidities. This method is suggested mainly for multiple-girder straight bridges and curved bridges with high torsional rigidity. However, the Canadian Highway Bridge Design Code (CHBDC 2000) [21], has recommended to use this method only for the analysis of straight box-girder bridges of multi-spine cross section but not multi-cell cross section.

2.3.2 Grillage Analogy Method
Grillage analysis has been applied to multiple cell boxes with vertical and sloping webs and voided slabs, see Fig. (2.1). In this method, the bridge deck structure may be investigated as an assembly of elastic structural members connected together at discrete nodes, i.e. the bridge deck is idealized as a grid assembly. Using the stiffness method of structural analysis as a primary approach makes it possible to analyze the structure.

Husain, (1964) [71], was among the first in idealizing a cellular plate structure as a grillage. Sawko and Willcock, (1967) [108], extended Husain's method to analyze cellular plate structure having variable section properties and several transverse webs. Sawko, (1968) [109], appeared the first to propose the effects of transverse cell distortion in cellular plate structure. Hook and Richmond, (1970) [65], used the Lattice Analogy (which models membrane and plate bending of structures as a lattice framework) in incorporating transverse beams for the analysis of cellular bridge decks. Hambly and Pennells, (1975) [57], applied this idealization to the multicellular superstructure and Kissane and Beal, (1975) [113], to curved multi-spine box-girder bridges. The continuous curved bridge is modeled as a system of discrete curved longitudinal members intersecting orthogonally with transverse grillage members. As a result of the fall-off in stress at points remote from webs due to shear lag, the slab width is replaced by a reduced effective width over which the stress is assumed to be uniform. The equivalent stiffness of the continuum is lumped orthogonally along the grillage members. Evans and Shanmugam, (1979) [48], applied the grillage analogy to the analysis of steel cellular plate structure. In case where the spacing of webs of one direction is more than 1.5 times the spacing of webs is in the other direction. Mohammed, (1994) [91], used the grillage method in the analysis of rectangular steel cellular plate structures using various types of meshes. Mohsin, (1995) [92], studied in detail the case of steel cellular plate structure with non parallel webs and diaphragms which are usually used in aircraft wings and some bridge decks or approaches. Fairooz, (1997) [52], used the grillage analogy method in the analysis of steel cellular plate structures curved in plane. Mashal, (1997) [88], used the grillage to analyze the steel cellular plate structures in their non-linear range to investigate the ultimate (collapse) loading. The grillage analogy method, is widely used in analyzing the cellular steel plate structure with different geometry and status [11, 16 and 20].

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Canadian Highway Bridge Design Code (CHBDC 2000) [21], has limited the use of this method to the analysis of voided slab and box-girder bridges in which the number of cells or boxes is greater than two.

2.3.3 Folded Plate Method
This method produces solutions for linear elastic analysis of a box girder bridge, within the scope of the assumptions of the elasticity theory. In this method a box girder bridge can be modeled as a folded system which consists of an assembly of longitudinal plate elements interconnected at joints along their longitudinal edges and simply-supported at both ends by diaphragms. These diaphragms are infinitely stiff in their own planes but perfectly flexible perpendicular to their own plane, see Fig. (2.2).

Figure (2.2): Description of Folded Plate Method.

A cellular deck may be considered as a particular type of folded plate structure. An accepted definition of a folded plate is a prismatic shell formed by a series of adjoining thin plate slabs rigidly connected along their common edges. The shell is usually closed and it ends with integral diaphragms, constructed conventionally of reinforced or prestressed concrete and the structure may be simply supported or continuous over several spans. Scordelis, (1960) [113], developed an analytical procedure for determining longitudinal stresses, transverse moments and vertical deflections in folded plate structures by utilizing matrix algebra. The procedure can be easily programmed for digital computers. Then, the method is based on the elasticity analysis of Goldberg and Levy, (1975) as developed by De Fries-Skene and Scordelis, (1964) [42].
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Marsh and Taylor, (1990) [87], developed a method that incorporates a classical folded plate analysis of an assemblage of orthotropic or isotropic plates to form box girders. One of the major drawbacks of the folded plate method is that it is tedious and complicated. It is noted that, the field of application of the folded plate method is restricted to right cellular bridge decks of uniform cross-section. According to the Canadian Highway Bridge Design Code (CHBDC 2000) [21], the applicability of this method is restricted to bridges with support conditions that are closely equivalent to line supports at both ends and also line intermediate supports in the case of multi-span bridges.

2.3.4 The Equivalent Sandwich Plate Method
Arendts and Sanders, (1970) [16], developed a method based on the concept of replacing the actual structure by an equivalent uniform plate. This method represents the top and bottom flange in the actual cellular deck by the flange in the equivalent sandwich plate and the webs and diaphragms in the actual cellular deck by a core material in the equivalent and sandwich plate. The assumed stress distribution in a plate element is assumed as follows: The normal and horizontal shear stresses are to be carried by the flanges only, while the vertical or transverse shearing stresses are carried by the core medium.

2.3.5 Finite Strip Method
The finite strip method can be regarded as a special form of displacement formulation of finite element method, see Fig. (2.3). Using a strain-displacement relationship, the strain energy of the structure and the potential energy of external loads can be expressed by displacement parameters. It employs the minimum potential energy theorem where at equilibrium; the values of the displacement parameters should make the total potential energy of the structure becomes minimal.

The finite strip method was introduced by Cheung, (1968) [31], and then Cheung and Cheung, (1971) [26], applied this method to analyze curved box girders. They programmed this method and used the program to solve the numerical examples of curved bridges as well as straight bridges by making the radius of curvature very large and the subtended angle very small. Kabir and Scordelis, (1974) [73], developed a finite strip computer program to analyze curved continuous span cellular bridges, with interior radial diaphragms, on supporting planar frame bents. At the same time Cusens and Loo, (1974) [39], presented a general finite-strip technique to single and multi-span box bridges with an extension to the analysis of prestressing forces induced prestressing cables in multi-span bridges. Cheung, (1984) [29], used a numerical technique based on the finite-strip method and the force method for the analysis of continuous curved multi-cell box-girder bridges. Ho et. al., (1989) [25], used the finite strip to analyze three different types of simply supported highway bridges, slab-on-girder, two-cell box girder, and rectangular voided slab bridges. Cheung and Li, (1989) [30], extended the applicability of finite-strip method to analyze continuous haunched box-girder bridges (with variable depth web strip). Although the finite strip method has broader applicability as compared to folded plate method, however the drawback is that the Canadian Highway Bridge Design Code (CHBDC 2000) [21], restricts its applicability to simply supported prismatic structures with simple line support.

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Literature Review

2.4 Methods of Idealization Used in This Work
There are two methods of idealization that are considered in the present work; one of these methods is the proposed method which is called as Frame Analogy Method that previously developed and modified by several authors and modified in this study and named the Panel Element Method (PEM) and the other method is the Finite Element Method (FEM) which is used for verification purposes.

2.5 Review of the Analogous Frame Approach
The analogous frame idealization procedure is an approach that is used to model core systems of flat sides and rectangular cross-section (closed or open). Moreover, this approach is usually used for static analysis only. A core system, in this approach, is idealized as an assemblage of a variety of one-dimensional frame elements. Each individual wall component is considered separately, with compatibility satisfied at junctions. Different studies have developed different ways of modeling the planer wall units, leading to a wide range of element types as seen hereafter. The analogous frame approach was originally used to analyze coupled planar wall structures [85]. It has been extended [60, 123] to deal with three-dimensional coupled non-planer wall assemblies. Macleod and Hosny, (1977) [84], idealized a core system as an assemblage of interconnected planar wall units, with warping displacements of non-planar walls evaluated as an integral part of the solution. Fig. (2.4) shows the planar wall element and the special bracing required to maintain rigid joint behavior as proposed by Macleod, where the wall elements are just normal column elements. Stafford Smith and Abate, (1984) [119], also adopted this idealization procedure in modeling non-planar walls, but with shear deformation be in walls included and the analogous frame models can be used like finite elements. Subsequently, Stafford Smith and Girgis, (1980) [120], developed two alternative analogous frame modules, namely, braced wide column analogy and braced frame analogy.

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Literature Review

(a): Wide column frame elements for coupled.

(b): Solid wall elements (arrows designate degrees of freedom).

(c): Perforated wall elements.

(d): Core cross-section at floor level.

Figure (2.4): Wall element used by Macleod and Hosny [84].

In the braced wide column analogy (Fig. 2.5), a similar module to the wide column analogy is used, but with diagonal bracing. The bracing are added to avoid coupling between the shear and the bending behavior in the wide column analogy. Therefore, a single module consists of rigid horizontal beams (or links) equal in length to the width of wall segment, connected by a single column. Hinged ends diagonal bracing are connecting the ends of these beams. A single rigid beam replaces adjacent beams of vertically adjacent modules in the frame and, joints between horizontally adjacent beams are hinged as shown in Fig. (2.6).

In the braced frame analogy, the module is symmetrical with a column at each side connected to rigid beams at top and bottom, with diagonal braces as shown in Fig. (2.7a). This might work satisfactorily as wall width unit for plane walls and for orthogonal assemblies of cores with not more than two walls connected at each corner. In general cases of cores subdivided into fractional wall-width modules and multi-wall assemblies, the available degrees of freedom at node would not allow the column of two adjacent modules to rotate independently, as they should. Consequently, the modules were made symmetric (Fig. 2.7b) with column connected to the rigid beams on left hand side, and hinged end link on the right side to release rotation.
22

Both the above idealization methods may be necessary for static analysis of cores subjected to lateral loads. However, for dynamic or earthquake response, these methods are too complicated and result in too many degrees of freedom required to adequately idealize a core system and do not fit to use in modeling a nonrectangular wall component. Kwan, (1991) [80], developed another element in which, the effect of wall shear deformation is allowed in the rigid arm rather than in the flexible column and, the rotational degree of freedom is defined as the rotation of the vertical fibers as shown in Fig. (2.8).

The methods based on the analogous frame approach which are currently available have several drawbacks in addition to those mentioned earlier, for example; (i) they mostly result in ill-conditioned property matrices due to the use of very rigid bracing elements; (ii) too many different types of elements are required to model the core system and (iii) they do not consider the out-of-plane bending behavior. Accordingly, Ali, (1993) [7], proposes another wall element to module non-planar core systems composed of flat wall units. In this wall element, local torsional stiffness, in-plane and out-plane shear deformation were included in the formulation. The in plane transitional degree of freedom (in plane of wall segment) was defined along the rigid arm, but, out-plane flexural of individual wall units was not considered.
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Literature Review

AL-Dolaimy, (1998) [5], modified Ali's element [7]. He used the developed wall element to idealize silos of circular and elliptic cross-section. The silo is idealized as an assemblage of solid planar wall elements that are developed on the basis of the wide column analogy, with compatibility requirements at the junctions of the components satisfied automatically in this idealization. Al_Juwary, (1999) [4], used new idealization to analyze straight cellular bridges with rectangular cross-section. Finally, Al_Khazraji, in (2010) [9], also used new idealization to analyze curved cellular bridges decks with trapezoidal cross-section.

2.6 General Behavior of Curved Box Girders
A general loading on a box girder, such as that shown in (Fig. 2.9a) for single cell box, has components which bend, twist, and deform the cross section. In a cellular bridge deck the structural behavior or the structural action is demonstrated by the type of the analytical solution technique. Fig. (2.10) shows some of the structural actions. The various factors shown in Fig. (2.10) are as follows: 1. Distortion or deformation of the cross section arising from the in-plane displacements of points on the cross section; this type of action can lead to distortional warping. Resistance to distortion is provided either by transverse diaphragms or, more normally for concrete, by increasing the bending strength of the walls of the box (see Fig. 2.10a). 2. Warping of the cross-section due to torsion, corresponding to out-of-plane or axial displacements of points on the cross-section, causing plane section not to remain plane (see Fig. 2.10b); in such a case, no distortion occurs. 3. Shear Lag; It describes the effect of the shear deformation on distributing the bending stresses in a box beam; it allows the structure to resist higher ultimate bending moment than that which is calculated by simple bending theory. When a box beam is subjected to torsion, a cross- section tends to warp from its original plane. If one end is restrained against warping, axial stresses are introduced and the shear flow is redistributed near the fixed end, this is also an effect of shear deformation and is sometimes called a Shear Lag effect, (see Fig. 2.10c).
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Literature Review

(a): Loading Components.

(b): Bending Load Actions.

(c): Torsional Load Actions.

Figure (2.9): General behavior of an open box section under gravity load showing separate effect. [134]

(a): Distortion or Deformation of cross-section.

(b): Torsional Warping of cross-section.

(c): Shear lag in bending.

Figure (2.10): Structural action of cellular structures. [106]

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Literature Review

Horizontally curved box girders applicable to both simple and continuous spans are used for grade-separation and elevated bridges where the structure must coincide with the curved roadway alignment. This condition occurs frequently at urban crossings and interchanges and also at rural intersections where the structure must conform to the geometric requirements of the highway. The objective of this section of the material is to present the overview of the general behavior of the box girder bridges. Horizontally curved bridges will undergo bending and associated shear stresses as well as torsional stresses because of the horizontal curvature even if they are only subjected to their own gravitational load. Fig. (2.9) shows the general behavior of an open box section under gravity load showing separate effect. An arbitrary line load on a simple span box girder (Fig. 2.9a) contains bending and torsional load components that have corresponding bending and torsional effects.

2.7 Free Vibration Analysis of Bridges
The free vibration analysis of planar curved beams, arches and rings have been the subject of numerous studies due to their wide variety of potential applications, such as bridges, aircraft structures, and etc. These structures are modeled as either extensional (including the extension of the neutral axis) or in extensional (neglecting the extension of the neutral axis), with Euler!Bernoulli and Timoshenko curved beams having been formulated for each model. Many methods have been employed to study the free vibration of curved members. Among the earliest studies of the dynamic behavior of single span constant thickness bridges [27], an attention must be made to the investigations of Inglis, Hilleborg, Biggs and others; Mise and Kunii, in which the bridge is assumed to behave like a beam. The assumption is valid only for bridges with long and heavy girders as are usual in railway plate girder bridge practice. Many studies, such as those of Rao and Sundararajan, and Tufekci and Arpaci, solve the equations of motion governing the in-plane vibration for classical boundary conditions. Shore and Chaudhuri, (1972) [116], studied the free vibration of horizontally curved beams using closed-form solutions of the equations of motion.
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Literature Review

Huffington and Hoppmann, (1958) [69], determined the exact frequencies and modal Eigen-function of rectangular orthotropic plates with "bridge-type" of boundary conditions by direct solution of the governing differential equation of motion using the Le'vy approach. Yamada and Veletsos, (1958) [132], have studied the vibration of single-span I-beam and slab bridge deck treating the slab as continuous beams over a series of flexible beams and using the Rayleigh-Ritz method. Based on the energy method they have also solved the free vibration problem by idealizing the structure as an equivalent orthotropic plate. Huang and Walker, (1960) [68], have compared the two approaches for five and nine beam bridges with edge beams. Chenug et. al., (1971) [27], used the Finite Strip Approach in the free vibration analysis of certain single and continuous span bridges for both constant and variable thickness with isotropic or orthotropic properties. Using a transfer matrix approach, Bickford and Strom, (1975) [19], obtained the natural frequencies and mode shapes for both the in-plane and out-of-plane vibrations of plane curved beams accounting for shear deformation, rotary inertia and extension of the neutral axis. Rasheed, (1998) [99], used the discrete element idealization scheme (space frame element) for static and dynamic (earthquake response) analysis of cable stayed bridges. Bridges of harp type cables are investigated under different types of static loading and due to the different directions of earthquake base excitations. Warping is considered as a seventh degree of freedom in the discrete element idealization (space frame element). Mahmood, (1999) [86], introduced a new development for the dynamic analysis of free vertical and lateral vibrations for self-anchored suspension bridges. He derived a governing differential equation of free vertical vibration of selfanchored suspension bridges. Finite element approach was also used in this study to solve the problem of free lateral vibration of suspension bridges. Howson and Jemah, (1999) [66], obtained the exact out-of-plane frequencies of curved Timoshenko beams using dynamic stiffness matrix, and discussed the effects of shear deflection and rotary inertia.
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Chapter Three

Finite Element Modeling

CHAPTER THREE

FINITE ELEMENT MODELING
3.1 General

There

are many methods that are available for analyzing curved bridges,

as mentioned earlier in Chapter 2. However, of all the available analysis methods, the finite element method is considered to be the most powerful, versatile and flexible method. The method of finite element can be used successfully for the analysis of a wide variety of highway bridges such as slab-type bridges, beam slab bridges, box girder bridges, and curved bridges. It is still the most general and comprehensive technique for static and dynamic analyses, capturing all aspects affecting the structural response. The other methods proved to be adequate but limited in scope and applicability. Due to recent development in computer technology, the finite element method has become an important part of engineering analysis and design because nowadays finite element computer programs are used practically in all branches of engineering. So, it is presented for the analysis of bridge decks treated as shell-type structures. In the current research, various structural elements are modeled using finite element method. In this chapter, a general description of the finite element approach is presented next followed by background information pertaining to the finite element program ANSYS [15], within the CivilFEM program for ANSYS that is utilized throughout this study for the structural modeling and analysis, and a cellular bridge deck of a curved and rectangular cross-section, and trapezoidal cross-section typically consists of planar units or non-planar units interconnected to each other to form a three-dimensional structural system. The Finite Element Idealization procedure, as used in this study for validation purpose, is also presented. Finally the description of the models of the curved box bridges is presented.
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Finite Element Modeling

3.2 The Finite Element Method (FEM)
The finite element method (FEM) is a numerical procedure for solving problems in engineering and mathematical physics, and it is the dominant discretization technique in structural mechanics. This numerical method of analysis starts by discretizing a model. So the basic concept in the physical FEM is the subdivision of the mathematical model into disjoint (non-overlapping) components of simple geometry called finite elements or elements for short. The discretization is the process where a body is divided into an equivalent system of smaller bodies or units called elements. These elements are interconnected with each other by means of certain points called nodes. The response of each element is expressed in terms of a finite number of degrees of freedom characterized as the value of an unknown function, or functions, at a set of nodal points. In structural problems, the solution is typically concerned with determining stresses and displacements. The finite element model gives approximate values of the unknowns at discrete number of points in a continuum. In the case of small displacements and linear material response, using a displacement formulation, the stiffness matrix of each element is derived and the global stiffness matrix of the entire structure can be formulated by assembling the stiffness matrices of all elements using direct stiffness method. This global stiffness matrix, along with the given displacement boundary conditions and applied loads is then solved, thus the displacements and stresses for the entire system are determined. The global stiffness matrix represents the nodal force-displacement relationships and can be expressed by the following equilibrium equation in matrix form:

3.3 The Finite Element Program: ANSYS
The finite element modeling and analysis performed in this study were done using a general purpose, multi-discipline finite element program, ANSYS [15]. ANSYS is a commercial finite element program developed by Swanson Analysis Systems, Inc. (SAS IP Inc 12th edition). The program is available for both PC and UNIX based systems. The analyses presented in this thesis were performed using ANSYS version 12.0. ANSYS has an extensive library of truss, beam, shell and solid elements. The elements that are used in this method of modeling are as follows: 1. Shell 63 (elastic shell): 2. Beam 4 (3-D Elastic Beam): A brief description of the elements used in the model is presented below:

3.3.1 SHELL63 (Elastic Shell) Element Description
SHELL63 is suitable for analyzing thin to moderately-thick shell structures. It is a four nodes element that has both bending and membrane capabilities. Both in-plane and normal loads are permitted. The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities are included in the element. A consistent tangent stiffness matrix option is available for use in large deflection (finite rotation) analyses. It is stated in ANSYS manual that an assemblage of this flat shell element can produce good results for a curved shell surface provided that each flat element does not extend more than a 156 arc. See SHELL63 in the Theory Reference for the Mechanical APDL (ANSYS Parametric Design Language), and Mechanical Applications of ANSYS program for more details about this element [15]. Fig. (3.1) below, shows the typical of shell element, nodal displacements, and the geometry of the SHELL63 (Elastic Shell) element.

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(a) Typical Shell Element.

(b) Nodal Displacements.

(c) SHELL63 (Elastic Shell) Geometry.

Figure (3.1): SHELL63 (Elastic Shell) Element. [15]

3.3.1.1 SHELL63 Assumptions
The important assumptions of the Shell63 element are as follows: i. Zero area elements are not allowed. This occurs most often whenever the elements are not numbered properly. ii. Zero thickness elements or elements tapering down to a zero thickness at any corner are not allowed. iii. The applied transverse thermal gradient is assumed to vary linearly with the thickness and vary bilinearly over the shell surface. iv. An assemblage of flat shell elements can produce a good approximation of a curved shell surface provided that each flat element does not extend more than a 15" arc. Shear deflection is not included in this thin-shell element.
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Finite Element Modeling

3.3.1.2 SHELL63 Input Data
The geometry, node locations, and the coordinate system for this element are shown in Fig. (3.1c): SHELL63 Geometry. The element is defined by four nodes, four thicknesses, and the orthotropic material properties. Orthotropic material directions correspond to the element coordinate directions. The element coordinate system orientation is as described in Fig. (3.1) above. The element x-axis may be rotated by an angle of THETA (in degrees). The thickness is assumed to vary smoothly over the area of the element, with the thickness input at the four nodes. If the element has a constant thickness, only one thickness need be input, and if the thickness is not constant, all four thicknesses must be input.

3.3.2 Beam4 (3-D Elastic Beam) Element Description
BEAM4 is a uni-axial element with tension, compression, torsion, and bending capabilities. The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. Stress stiffening and large deflection capabilities are included. The required inputs for this element are the cross-sectional properties such as, the moment of inertia, the cross-sectional area and the torsional properties. See BEAM4 in the Theory Reference for the Mechanical APDL and Mechanical Applications of ANSYS program for more details about this element [15]. Fig. (3.2) below, shows the typical of beam element, and the order of degrees of freedom of the BEAM4 (3-D Elastic Beam) element.

3.4 Case Studies of Curved Box-Girder Bridge
There are primarily four case studies of cross-sectional area of curved boxgirder bridges that are modeled in ANSYS for the current study. The description of all case studies are shown as follows:

3.5 Description of the Box-Girder Bridge Models 3.5.1 Curved Box Beam Model (MB)
A curved box beam bridge model is made using beam4 elements for the box girder bridge. Beam4 elements are used as well to model this type of bridge. The cross-section area of the beam bridge, length, thicknesses and the material properties are required input for beam4 element. The length of the curved bridge is (20 m) and (30 m) from support to support at center, and has a radius of curvature of (57.3 m), with angle of curvature of (206 degree). The models of this type of element are: cantilever model (MBC), simply supported ends model (MBS), with rectangular cross-section and single cellular. Typical cantilever curved box beam models (MB) with meshing by ANSYS, are shown in Figs. (3.8 and 3.9).

Figure (3.8): Typical Cantilever Curved Box Beam Model.

Figure (3.9): Typical Simply Supported Curved Box Beam Model. 37

Chapter Three

Finite Element Modeling

3.5.2 Curved Box Shell Model (MS)
A curved box shell bridge model is made using Shell63 elements for the box girder bridge. Shell63 elements are used as well to model this type of bridge. The cross-section area, length, thicknesses and the material properties of the beam bridge are required input for Shell63 element. The length of the curved bridge is (20 m) and (30 m) from support to support at center, and has a radius of curvature of (57.3 m), with angle of curvature (206 and 306 degree) respectively. The models of this type of element are as mentioned above: cantilever end model with rectangular cross-section and single cellular (MRSCS) and with double cellular (MRSCD), and trapezoidal cross-section with single cellular (MTSCS) and with double cellular (MTSCD), simply supported ends model with rectangular crosssection and single cellular with partial restrained (MRSSS-PR) and fully restrained (MRSSS-FR), and with double cellular with partial restrained (MRSSD-PR) and fully restrained (MRSSD-FR), and trapezoidal cross-section with single cellular with partial restrained (MTSSS-PR) and fully restrained (MTSSS-FR), and with double cellular with partial restrained (MTSSD-PR) and fully restrained (MTSSD-FR). Typical cantilever curved box shell models (MS) with meshing by ANSYS, are shown in Figs. (3.10, 3.11, 3.12 and 3.13).

3.6 Modal Analysis
Modal analysis is used to determine the modal properties and the vibration characteristics such as (natural frequencies and mode shapes) of a structure or a machine component from its specified geometrical and material properties. The natural frequencies and mode shapes are important parameters in the analysis and design of a structure for dynamic loading conditions. This is probably the most common type of dynamic analysis and is referred to as an Eigen-value analysis. This analysis is the undamped free vibration response of structure caused by an initial disturbance from the static equilibrium position. Normally, only the first few Eigen-values of the model are interesting. Indeed, since the finite element model is an approximation of the structure, then the higher Eigen-values and corresponding vectors are inaccurate. The theoretical solution implies that the structure will vibrate in any mode shape indefinitely. However# since there is always some damping present in any structure, the vibrations eventually decay. Obtaining the Eigen-value of free vibration from the finite element, ANSYS models can be utilized in understanding the curved box-girder bridge behavior. In addition, it can also be used to compare the mode-shape or the stress profiles. Therefore, creating the same general construction of curved bridge models with same boundary conditions is required. In modeling the bridges using ANSYS, a FEA model was created using the Graphical User Interface (GUI) other than command prompt line input. A modal analysis can be performed using the ANSYS solver, and there are many numerical algorithms to extract the Eigen matrices from the spatial model. Since these are well established and documented, the theory behind them is not presented here. For the purposes of this study, the Block Lanczos iteration method of Eigen solution, as implemented in commercial finite element software called ANSYS is utilized. The important material properties of ANSYS models (Linear, Elastic and Isotropic) that are used in the studied cases are shown in Table (A-1) in Appendix (A).

Today, bridge construction has achieved worldwide level of importance. Due to
efficient dissemination of congested traffic, economic considerations, and aesthetic desirability horizontally curved box girder bridges have become increasingly popular nowadays in modern highway systems, including urban interchanges. Single or multicell reinforced and pre-stressed concrete box-girder bridges have been widely used due to economic and aesthetic solution for overcrossing separation structures and viaducts and are found in today's modern highway systems [125]. A cellular curved bridge deck of a square or rectangular and trapezoidal in cross-section typically consists of planar units or non-planar units interconnected to each other to form a three-dimensional structural system. An idealization procedure for modeling cellular type bridge decks designated as the Panel Element Method (PEM) is proposed in this chapter. The Finite Element Method (FEM) idealization procedure, as used in this study for validation purpose, was also presented with details in the previous chapter 3. An equation of motion that governs the response of cellular bridge decks subjected to base excitations is also presented in this chapter. Finally the description of the models of the curved box bridges is presented.

4.2 Description Of The Panel Element Method (PEM)
An idealization procedure for panel element (PE), which is designated as the Panel Element Method (PEM) is presented here. This procedure may be classified as an analogous frame approach, in which the basic element used is obtained by modifying the element used in an existing analogous frame method.
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Chapter Four

Idealization of The Problem

The description of the Panel Element (PE), and the Panel Element Method (PEM) that are proposed as an idealization procedure for modeling the curved cellular type bridge decks, is as follows:

4.2.1 The Assumptions
The derivation and modification of the Panel Element Method (PEM) takes into consideration the following assumptions that are found to be necessary for the formulation of the problem: 1. The bridge is assumed to be as an assemblage of finite number of the flat plates or wall panels. 2. Each wall or slab panel is modeled by the conventional space frame element. 3. In-plane flexural and shear deformations of the individual panel element (PE) are considered. 4. Out-of-plane flexural and shear deformations of each individual panel element (PE) are also considered. 5. Diaphragms are considered to be infinitely rigid in their own planes and flexible out-of-plane. 6. A partially rigid interface is assumed between panel elements and diaphragm in both directions, that is; relative rotational, is allowed between the panel element and the diaphragm only in the plane of panels with no distortion of the cross-section

4.2.2 The Basis of the Panel Element
The proposed panel element is based on the Wide Column Analogy shown in Fig (4.1a), where each element is considered to consist of two horizontal rigid arms, each of length equal to the width of the panel segment under consideration. These rigid arms are centrally connected by a central flexible beam of length equal to the length of the panel unit. The element is assumed to have four nodes, two of which to be located at either end of each rigid arm to maintain the compatibility with the other panel components (see Fig. (4.1c)).
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Chapter Four

Idealization of The Problem

Three orthogonal translational degrees of freedom (d.o.f) in addition to one out-of-plane rotational degrees of freedom are assumed at each nodal point. The rigid arm assumption, however, constrains the horizontal in-plane translation degrees of freedom along the axis of the rigid arm and the out-of-plane rotational degrees of freedom at the two nodes on each rigid arm to be identical. Therefore, each Panel element (PE) has twelve degrees of freedom (d.o.f), as shown in Fig. (4.1d).

(a) Planer Panel Unit of a Deck

(b) Standard Space Frame Element

(c) Wide Column Analogy

(d) Panel Element (PE)

Figure (4.1): Development of The Panel Element (PE). [9]

The interpretation of these twelve degrees of freedom can be understood as follows: At each side of the panel element (PE) end, or at each position of a diaphragms, the two longitudinal translations (in the horizontal direction and along the longitudinal axis of the deck) account for axial and in-plane rotation of the panel element at that side or position. The description of the panel element (PE) in-plane of paper and the order of all degrees of freedom of it, is shown in Fig. (4.2) below.
43

The two out-of-plane degrees of freedom (d.o.f) account for torsion and transverse shear deformation. The in-plane lateral degree of freedom accounts for the in-plane flexural and shear deformation. Finally, the remaining out-of-plane rotation accounts for out-of-plane flexural behavior of the panel unit. The description of the panel element (PE) out-of-plane of paper and the order of all degrees of freedom of it, is shown in Fig. (4.3).

4.2.3 Stiffness Matrix of the Panel Element
The element stiffness matrix [Ke] corresponding to the degrees of freedom (d.o.f) of the in-plane of paper and out-of-plane of paper that are shown in Fig. (4.2) and Fig. (4.3) above, is derived from the stiffness matrix [Kc] [15] of the standard space frame element of Fig. (4.1b) above; as the degrees of freedom of the two elements are kinematically relatable, such that:

Where, D = The width of the panel unit element. The stiffness matrix of the panel element [Ke], therefore, may be given by:

[Ke] = [Te]T. [Kc] . [Te]
Where,

……………………………………. (4.5)

[Ke] = The stiffness matrix of the panel element. [Te]T = The transpose matrix of [Te] matrix. [Kc] = The stiffness matrix of the standard space frame element. The stiffness matrix [Kc] is the stiffness matrix of the standard space frame element that is shown in Fig. (4.1b), which can be written in the form:

é ù [K c ] = ê [K ii ]T [K ij ]ú ë[K ij ] [K jj ]û
Where,

……….………………………………. (4.6)

[Kii] = The stiffness of sub-matrix of panel element corresponding to d.o.f at node (i). [Kij] = The stiffness of sub-matrix of panel element corresponding to forces at node (i) due to displacements at node (j). [Kij]T= The transpose matrix of [Kij] matrix. [Kjj] = The stiffness sub-matrix of panel element corresponding to d.o.f at node (j).

A = The cross-sectional area of the panel element, and it is equal to:

A=D.t
Where,

……………………………………. (4.11)

t = The thickness of the panel element. h = The length of the panel element along the longitudinal axis of the deck. Ix = The moment of inertia of the panel cross-section about the local x-axis, and it is given by:

D ×t3 Ix = 12
Also,

……….………………………………. (4.12)

Iy = The moment of inertia of the panel cross-section about the local y-axis, and it is given by:

t × D3 Iy = 12
And,

……….………………………………. (4.13)

E = The modulus of elasticity (Young's modulus) of the panel unit material. G = The shear modulus of the panel unit material, and it is given by:

G =
Where,

E 2 (1 + u )

……….………………………………. (4.14)

! = The Poisson's ratio of panel element material, and the typical Poisson's
ratio for the concrete material is generally taken equal to (0.1~0.2). And, J = The torsional moment of inertia of the panel element, and it is equal to:

J = Jz
Or,

( if Iz = 0 ) ( if Iz 7 0 )

…………………………. (4.15a) …………………………. (4.15b)

J = Iz
Where,

Jz = The polar moment of inertia of the panel element about the local z-axis, and it is given by:

Jz = Ix + Iy

……………………………………. (4.16)
48

Chapter Four Where,

Idealization of The Problem

Iz = The torsional moment constant of inertia of the panel element about z-axis, and it is given approximately as shown in Table (A-3) in Appendix (A), for the rectangular cross-section area equal to: i- For solid rectangular sections:

Where, t = The thickness of cross-section of the deck. b = The width of the cross-section of the deck. d = The height of the cross-section of the deck. tw = The web thickness of the cross-section of the deck. ts = The slab thickness of the cross-section of the deck. And, the k parameters are given by:

#x = The shear factor of the panel element cross-section in local x-direction, and it is given by:

F # x=
And,

12 E × I y
2 G× A e ×h

…………………………….……. (4.25)

#y = The shear factor of the panel element cross-section in local y-direction, and it is given by:

F # y=
Where,

12E × I x 2 G× A e ×h

…………………………….……. (4.26)

Ae = The effective shear area of the panel element cross-section that is normal to the z-direction, and it is given by:

Ae = A / 1.2

……………………………………. (4.27)

4.2.4 Mass Matrix of the Panel Element
The mass matrix of the panel element can be also expressed in terms of the space frame unit, as follows:

4.2.4.1 Consistent Mass Matrix of the Panel Element
The element mass matrix [Me] corresponding to the degrees of freedom (d.o.f) of the in-plane of paper and out-of-plane of paper that are defined at the nodal points of the panel element shown in Fig. (4.2) and Fig. (4.3) above, is also derived from the mass matrix of the standard space frame element of Fig. (4.1b) above; the mass matrix [Mc] [15] of the standard space frame element correspond to the kinematically equivalent degrees of freedom (d.o.f) that are defined in Fig. (4.1b) as follows: The mass matrix of the panel element [Me], therefore, may be given by:

[Me] = [Te]T. [Mc] . [Te]
Or,

…………………………………. (4.28a) …………………………………. (4.28b)

[Me] = [Te]T. [Ml] . [Te]

50

Chapter Four Where, [Me] = The mass matrix of the panel element.

Idealization of The Problem

[Te] = The transformation matrix of the panel element, defined in Eq. (4.2). [Te]T = The transpose matrix of [Te] matrix. [Mc] = The consistent mass matrix of the standard space frame element. [Ml] = The lumping mass matrix of the standard space frame element. The transformation matrix of the panel element [Te] represents a kinematics transformation matrix defined in Eq. (4.2) and Eq. (4.3) for the panel element in-plane of paper, and Eq. (4.2) and Eq. (4.4) for the panel elements out-of-plane of paper. The mass matrix [Mc] is the mass matrix of the standard space frame element that is shown in Fig. (4.1b), which can be written in the form:

# = The mass density of the panel unit material.
rg = The radius of gyration of the panel cross-section about the local z-axis, and it is given by:

rg =

Jz A

….………………………………. (4.34)

4.2.4.2 HRZ Mass Matrix of The Panel Element a. HRZ Mass Matrix Lumping Scheme
The name "HRZ" identifies the authors of the procedure that shown in (b). The HRZ Lumping Scheme [35] of the mass matrix is an effective method for providing a diagonal mass matrix. It can be recommended for arbitrary elements. The essential idea is to compute only diagonal terms of the consistent element mass
52

Chapter Four

Idealization of The Problem

matrix, which must be scale them so as to preserve the total element mass, i.e. it must to scale them in such a way that the total mass of the element is preserved. The result is a diagonal mass matrix. Recognizing that there may be both translational and rotational degrees of freedom (d.o.f), which may describe motion in one, two, or three coordinate directions, the following steps might be followed for an element of total mass (mp).

b. HRZ Lumping Procedure
Specifically, the procedural steps are as follows: 1. Compute only the diagonal coefficients (mii) of the consistent element mass matrix; i.e. (m11, m22, ….., mii, ….., mnn), see Fig. (4.4). 2. Compute the total mass of the panel element, (mp); and it is equal to:

mp = # . A . h
Where,

……………………………………. (4.35)

mp = The total mass of the panel element. 3. Determine a number (Sc) for each coordinate direction in which motion is described by the element degrees of freedom (d.o.f); by adding the diagonal coefficients (mii) associated with translational degrees of freedom (d.o.f) (but not rotational degrees of freedom (d.o.f), if any) that are mutually parallel and in the same direction. The (Sc) number is equal to:

Sc = mu1 + ..... + mu i + muj + ..... + mu n -ndofs +1
Or,

……………. (4.36a) ……………. (4.36b)

Sc = mv1 + ..... + mvi + mvj + ..... + mvn - ndofs + 2
Where,

ndofs = The Number of degrees of freedom (d.o.f) per one section (segment). 4. Scale all diagonal coefficients by multiplying all coefficients (mii) that are associated with this direction by the ratio (mp/Sc), thus preserving the total mass of the element. 5. Applying the above steps to the consistent mass matrix of the panel element to convert it to a simple lumped mass matrix as shown below:
53

7. And from the all consistent mass matrices of the panels elements, the HRZ lumping is obtained. The HRZ lumping technique provides a simple diagonal mass matrix but no reduction in the total number of degrees of freedom (d.o.f) of the global structure and simplifies the numerical solution due to the low storage capacity and reduction of the elapsed time.

54

Chapter Four

Idealization of The Problem

(a) Typical Bridge Deck

(b) Idealized Deck Cell

Figure (4.4): Idealization of the Panel Element (PE). [4]

4.2.5 Effect of The Rigid Diaphragms
In the most known cases, all opposite corners of the bridge cell are, generally, connected using a wall or a beam which is assumed to be infinitely rigid within its plane or longitudinal axis, respectively, and hence named "Rigid Diaphragms". These rigid diaphragms could be located at right, left or at either side of the panel elements that are resembled by a component element.
55

Chapter Four

Idealization of The Problem

The advantage of the transverse member or the rigid diaphragm is to distribute the load transversally across the width of the deck and to resist the effect of distortion on the bridge cross-section. These rigid diaphragms are assumed to be infinitely rigid in their own plane, and completely flexible out of plane. In the proposed Panel Element Method (PEM), all units are assumed to be connected through diaphragms on both sides or on one side at the left or at right. Therefore, all the panel degrees of freedom in the plane of the rigid diaphragm, (see Fig. 4.5a & 4.5c), are kinematically dependent on the three in-plane degrees of freedom (d.o.f); (two translations and a rotation) of a master node, that is defined in this study at the center of mass of the curved cellular bridge cross-section, as shown in Fig. (4.5b and 4.5d) for out-of-plane paper.

This assumption reduces the number of dynamic structure degrees of freedom, with the tacit assumption that the cellular curved bridge cross section does not distort, which is reasonable for most cellular curved bridges. This tacit assumption means; for the bridge deck with an intermediate diaphragm at the certain locations, the cross-section is not distorted, while all translational structure degrees of freedom along the longitudinal axis of the deck are not constrained, thus, and allowing longitudinal cross-section warping to occur when the bridge undergoes torsional deformation. As a result of this transformation, the contribution of the panel element stiffness and mass matrices [Kme] and [Mme], respectively, to the structure property matrices with respect to the degrees of freedom (d.o.f) of a typical cell panel (individual or component elements of a cell shape) shown in Fig. (4.6 and 4.7), will take the form shown below, i.e. the stiffness matrix of the panel element [Kme] within effect of the rigid diaphragms, will take the form:

[Kme] = [Tg]T. [Ke] . [Tg]
Where,

…………………………………. (4.38)

[Kme] = The stiffness matrix of the panel element within the effect of diaphragm. [Tg] = The transformation matrix of the constrained within diaphragms. [Tg]T = The transpose matrix of [Tg] matrix. [Ke] = The stiffness matrix of the panel element, Eq. (4.5).

Also, the mass matrix of the panel element [Mme] within the effect of the rigid diaphragms, will take the form:

[Mme] = [Tg]T. [Me] . [Tg]

…………………………………. (4.39)

Where, [Mme] = The mass matrix of the panel element within the effect of diaphragm. [Me] = The mass matrix of the panel element, Eq. (4.28).
57

Chapter Four

Idealization of The Problem

The transformation matrix [Tg] is a transformation matrix relating the panel element degrees of freedom of each individual element to the global component element degrees of freedom (d.o.f) of the structure, shown in Fig. (4.5) and Fig. (4.6b), and it is given by:

The transformation matrix [Tg] takes one of the forms below: 1) For the cases of panel elements constrained by rigid diaphragms at both ends:

[T ] [0 ] ù [T ] = é ê [0 ] [T ]ú ë û
g1 g g1

…………………………………. (4.41a)

Or, 2) For the cases of panel elements constrained by rigid diaphragms at their left ends:

[T ] [0 ]ù [T ] = é ê [0 ] [1]ú ë û
g1 g

…………………………………. (4.41b)

Or, 3) For the cases of panel elements constrained by rigid diaphragms at their right ends:

[1] [0 ] ù [T ] = é ê [0 ] [T ]ú ë û
g g1

…………………………………. (4.41c)

Or, 4) For the cases of panel elements with no such constrained with any rigid diaphragms at any of their sides, then no transformation is needed. So, the transformation matrix can be written as follows:

xl = The horizontal dimension between the master node and the far left node for both the in-plane and out-of-plane panel elements. xr = The horizontal dimension between the master node and the near right node for both the in-plane and out-of-plane panel elements. yb = The vertical dimension between the master node and the bottom node for both the in-plane and out-of-plane panel elements. yt = The vertical dimension between the master node and the top node for both the in-plane and out-of-plane panel elements. The master node is the center of mass (C.G.) of the curved cellular bridge cross-section, as shown in the figures below.

4.2.6 Coordinate Systems
Two types of coordinate systems are used in this study, as shown in Fig. (4.13), and these two coordinate systems are defined as follows:

4.2.6.1 Global Cartesian Coordinate System (X, Y and Z)
The global Cartesian coordinate system (X, Y and Z) is used to define the nodal coordinates and

displacements.

Figure (4.13): Global and Local Cartesian Coordinate System. [135]

4.2.6.2 Local Cartesian Coordinate System (x', y' and z')
The local Cartesian coordinate system is used to define local stresses and strains at any point within the shell element. At such a point the z' direction is taken to be normal to the surface of the local x' and y', where z' is a constant. The center of the local Cartesian coordinate system coincides with the center of element coordinates, Fig. (4.14) below. The local coordinate system varies throughout the complex structure like the curved box-girder bridges, so it is useful to define the direction cosine matrix, which enables transformation between the local and the global coordinate systems. The matrix same is transformation applicable to

4.2.7 3D-Coordinate Rotation Matrix
Rotation matrices are essential for understanding how to convert from one reference system to another. Converting from one reference system to another is essential for computing joint angles. This section provides a specific technique to reference causes and effects by 3d-coordinate systems and how to transform these 3d-coordinate systems from one system to another. These techniques will provide a basis for determining the global stiffness and the global mass matrices of a member and developing the joint stiffness and mass solution. The 3D transformation from a local coordinate to a global coordinate system can be performed as illustrated in Fig. (4.15) below, in which (x, y and z) is the global system and (x2, y2 and z2) often denoted by ( xyz), is the local system.

(a) 3D-Rotation Axes

(b) Projection Lengths

Figure (4.15): Transformation from local coordinate system to global Coordinate System. [75]

4.2.7.1 Transformations Into System Coordinates
The stiffness and mass matrices of the panel element refer to the element local axes (x',y' and z'). In general each element of the cellular curved bridge may be arbitrarily oriented in space, therefore, it is required to transform each panel element's stiffness and mass matrices to the global degrees of freedom before the assemblage of the system stiffness and mass matrices for the whole structure. The global axes (X, Y and Z) are brought to coincide with the local axes (x', y' and z'), by a sequence of rotations about (Y, X and Z) axes respectively, see Fig. (4.16) which shows the three rotations of the local axes [79].
65

Chapter Four

Idealization of The Problem

(a)

(b)

(c)

Figure (4.16): Rotation Transformation of Axes for 3-D System. [79]

4.2.7.2 Local To Global System Conversion
The element coordinates are related to the global coordinates by: [15]

{Ul} = [TR] . {u}
Where,

…………………………………. (4.48)

{Ul} = The vector of displacements in element Cartesian coordinates. [TR] = The 3d-coordinate transformation matrix of the panel element. {u} = The vector of displacements in global Cartesian coordinates. The 3d-coordinate transformation matrix of the panel element [TR], is used to convert the stiffness and mass matrices of the panel element from the local coordinate to the global coordinate system and it can be expressed as:

x1, y1 and z1 = The x, y and z coordinates of node no.1 of the panel element. x2, y2 and z2 = The x, y and z coordinates of node no.2 of the panel element. Lxy = The projection of length onto X-Y plane. "r = The user-selected adjustment angle of the rotation of each panel element in direction of longitudinal of the bridge. dt = The tolerance number, and it is equal to: …………………………………. (4.51g)
67

dt = 0.0001 h

Chapter Four

Idealization of The Problem

4.2.8 The Global Stiffness and Mass Matrices
The global stiffness matrix of the panel element [KGe] within the effect of the 3d-rotation System Conversion, will take the form:

[KGe] = [TR]T. [Kme] . [TR]
Where,

…………………………………. (4.52)

[KGe] = The global stiffness matrix of the panel element within conversion. [TR] = The transformation matrix of the 3d-rotation systems, Eq. (4.49). [TR]T = The transpose matrix of [TR] matrix. [Kme] = The local stiffness matrix of the panel element, Eq. (4.38).

Also, the global mass matrix of the panel element [MGe] within the effect of the 3d-rotation System Conversion, will take the form:

[MGe] = [TR]T. [Mme] . [TR]
Where,

…………………………………. (4.53)

[MGe] = The global mass matrix of the panel element within conversion. [Mme] = The local mass matrix of the panel element, Eq. (4.39).

4.2.9 Cellular Bridge Idealization Using (CEM)
The global degrees of freedom [U], for the typical cellular bridge deck system as shown in Fig (4.5, 4.7 and 4.17) for single and double cellular-bridge deck, respectively, by using the Component Element Method (CEM), are defined as follows:

{ U }= {

u

v

w

qz

q L}

……………………………. (4.54)

Where u, v and w denote the translational displacements in three orthogonal directions X, Y and Z, respectively; "z is the rotational displacement about the local longitudinal (Z) axis; and "L denotes the local rotational displacements about the minor axis of each individual panel element. Accordingly, the stiffness and mass matrices of the component element [KCG] and [MCG] will be obtained as shown below.
68

Chapter Four

Idealization of The Problem

4.2.10 The Component Stiffness and Mass Matrices
The global stiffness matrix of the component element [KCG], after assemblage of all the stiffness matrices of the panel elements, as shown in Fig. (4.7), will take the form:

Kuw 0 é Kuu ê Kvv Kvw ê Kww [KCG ] = ê ê ê Symmetric ê ë
Where,

0 Kvq Z 0 KJZq Z

Kuq L ù Kvq L ú ú 0 ú ú 0 ú KJLq L ú û

……………………. (4.55)

[KCG] = The global stiffness matrix of the component element. Also, the global mass matrix of the component element [MCG], after assemblage of all the mass matrices of the panel elements, as shown in Fig. (4.7), will take the form:

Muw 0 é Muu ê Mvv Mvw ê Mww [MCG ] = ê ê ê Symmetric ê ë
Where,

0 MvqZ 0 MJZqZ

MuqL ù MvqL ú ú 0 ú ……………………. (4.56) ú 0 ú MJLqL ú û

[MCG] = The global mass matrix of the component element. In Eq. (4.55) and Eq. (4.56), each of the sub-matrices Kuu, Kvv, K"z"z, Muu, Mvv and M"z"z is of order (np), where (np) is the number of panels that extend between two consecutive diaphragms. The sub-matrices Kww and Mww are of order (np x mp), where (mp) is the number of longitudinal displacements at each diaphragms position, that is equal to the number of nodes at the corresponding diaphragms. Also, the K"L"L and M"L"L are of order (qp x np), where (qp) is the number of panel elements that form and represent the cellular-bridge deck system.
69

Chapter Four
Out-of-plane of paper

Idealization of The Problem
In-plane of paper

(a): Local (d.o.f) of the Double Cell Structure.

(b): Local (d.o.f) of the In-Plane Panel Elements.

(c): Local (d.o.f) of the Out-of-Plane Panel Elements. In-plane of paper
e10,"L e11,"L

e3,"Z e14,"L e15,"L e16,"L

e12,"L

e13,"L

Out-of-plane of paper (d): Global (d.o.f) of the Component Element.

Figure (4.17): Idealization of The Double Cellular-Bridge Deck By Using The Component Element Method (CEM). 70

Chapter Four

Idealization of The Problem

4.2.11 Finite Element Idealization Method
The finite element idealization procedure is used in this study to validate the use of the proposed Panel Element Method (PEM) and for comparison purposes. All analysis using the finite element idealization method is done using ready software (ANSYS12), and (CivilFEM12) for (ANSYS12). A cellular bridge deck normally consists of top and bottom slabs, vertical webs and transverse members (rigid diaphragms), all these parts are modeled by an assemblage of the general four nodes flat shell element as shown in Fig. (4.18). The assumption of transverse members (rigid diaphragms) are rigid in its plane and flexible out of its plane restrains all the in-plane degrees of freedom (d.o.f) of those nodes of the transverse member from being slaved to a master in-plane degrees of freedom (d.o.f) defined at the center of mass of the cellular bridge deck cross-section. However, all degrees of freedom which are not in-plane of diaphragms are not constrained to this transformation. To ensure convergence and accuracy of the results, for free vibration and earthquake response analysis of the numerical examples studied, a certain mesh size is initially selected to model the structure, and subsequently, the mesh is made finer till a maximum difference of less than (2%) is reached in two successive solutions. This idealization method is presented with details in the chapter (3).

4.2.12 Governing Equation of Motion 4.2.12.1 Basic Excitation Characteristics
The governing equation of motion of a bridge deck subjected to an earthquake base excitation is based on the following assumptions: i. Base excitation is characterized by two components, each of which is transverse to the deck longitudinal axis, but, one is assumed to be in the vertical direction and other in the horizontal direction. ii. The same earthquake-based motion acts simultaneously on all parts of the structure's foundation; this is based on the assumption that the foundation-soil is relatively rigid, and the span of the bridge is small relative to the earthquake wave length in the foundation-soil medium. Since the cellular bridge deck is assumed to behave linearly in response to the earthquake-based excitation, only the peak response of the system is interesting. Therefore, modal combination scheme is used. Each earthquake excited component (in X and Y-direction) is characterized by the Al-Hindya smooth pseudo-acceleration design spectrum of Fig. (4.20) [5]. This spectrum can be divided into acceleration controlled (short vibration periods T), velocity-controlled (medium-period) and displacement-controlled (long-period) regions. The (5%) damping ratio is used to construct the spectrum. The pseudo acceleration is constant in part of the acceleration-controlled region, varies as (1/T) in the velocity-controlled region and as (1/T2) in the displacement-controlled region.

4.2.12.2 Formulation of The Governing Equation
For the typical cellular bridge deck system shown in Fig. (4.4), the governing equation of motion when subjected to a single component of uniform ground motion is given by [26]:

. [M] . {'} + [C] . {,} + [K] . {U} = - [M] . {R} . 'g

………. (4.57)

72

Chapter Four

Idealization of The Problem

In which [M], [C] and [K] denote, respectively, the mass, damping and stiffness matrices of the bridge deck, corresponding to the structure degrees of freedom (d.o.f) {U}; {R} denotes an earthquake influenced vector consisting, of ones and zeros, where ones correspond to the degrees of freedom in the direction of the earthquake-based excitation, and zeroes elsewhere. Generally, there are three types of damping mechanisms in the cementation materials, which are, solid (hysteresis), viscous and frictional damping. The first represents the energy absorbed by internal friction of material, and the damping force is proportional to displacement and depends on the elastic modulus of the material. The second mechanism has a damping force proportional to the velocity of motion and is, therefore, a function of displacement and frequency in an oscillating system. The frictional damping arises from dry sliding of surfaces and in this case the damping force is nearly constant. In the present study, the damping ratio is directly defined in each mode of vibration as a viscous damping ratio ( . ) which is assumed to be the same for all vibration modes of the structure.

4.2.13 Computer Programs By MATLAB
A computer program was written for the dynamic analysis of the cellular bridge decks by using the proposed Panel Element (PE) method. Dynamic analysis which has been adopted consists of free and forced vibration (earthquake response analysis). The program group of "FSPE-DYNAMIC" is coded in MATLAB language by using a PC-computer. This program is used to analyze the curved cellular bridge decks for any support condition and under any live load type. A complete description of the program "FSPE-DYNAMIC" structure is given in the flow chart in Fig (4.19), and the MATLAB program codes are in Appendix (C). The important material properties of MATLAB models (Linear, Elastic and Isotropic) that are used in the studied cases which are shown in Table (A-2) in Appendix (A).
73

The buildings and structures in civil engineering are subjected to static and
dynamic loadings. Vibrational analysis of buildings is important as they are continuously subjected to dynamic loads like wind, earthquake etc. Various classical methods are presented which could be used to solve these problems and nowadays many software have also come up which help one to predict the behavior of the structures more accurately. When a structure is subjected to an excitation (force is only required to initiate), it doesn't have any further role. Hence in an un-forced condition it vibrates freely and finally comes to rest due to damping. These vibrations are called FREE VIBRATIONS, and frequencies are called NATURAL FREQUENCIES of vibrations. Any building or structure can be modeled into frames. These frames can be analyzed, and the behavior of the structure can be predicted. Finite Element Method (FEM) is a versatile tool which can be used to mathematically model and analyze the structure. In this chapter, the proposed Panel Element idealization procedure (PE) is validated by the free vibration response analysis of cellular bridge decks in different configurations. The natural frequencies and the corresponding global mode shapes for different cases of cellular bridge decks were estimated in this chapter using the Panel Element Method (PEM), vis-a-vis, the Finite Element Method (FEM). Decks of different configurations are also studied to evaluate the range of applicability of the proposed idealization scheme in modeling the cellular bridge deck.
78

Chapter Five

Free Vibration Analysis

5.2 Damped Vibration System
The governing equation of motion for any damped system is given by [34]:

. [M] . {'} + [C] . {,} + [K] . {U} = {F(t)}

…..……. (5.1)

Where {'}, {,} and {U} are the time dependent displacement, velocity and acceleration vectors, respectively. On the other hand, [K], [C] and [M] are the global stiffness, damping and mass matrices of the system, respectively. {F(t)} is the applied load vector. The system is assumed to have classical damping, thus, the damping matrix in the present study is of the form:

5.3 Undamped Free Vibration Analysis
The governing equation of motion for an undamped free vibration system can be obtained by omitting the damping matrix and the load vector from Eq. (5.1) above, such that:

[M] . {'} + [K] . {U} = {0}
; in which: {0} is a zero vector.

……......................…. (5.3)

Based on the assumption that, the free vibration motion characteristic of the system is a simple case of a harmonic type, it is expressed as:

ˆ) {U(t)} = {*} . Sin (4i . t + q
Where,

……......................…. (5.4)

* : The mode shape of the system.
ˆ : The phase angle. q
This above equation can be expressed as follows:

U = 9 . Sin 4t
79

……......................…. (5.5)

Chapter Five

Free Vibration Analysis

where ( 9 ) represents the eigenvector and ( 4 ) is the natural frequency of the structure. Substituting ( U ) and ( ' ) of Eq. (5.5) into Eq. (5.3) yields:

K . 9 = 42 . M . 9
Or,

……......................…. (5.6) ……......................…. (5.7)

K.9=/.M.9
the natural frequencies vector ( 4 ).

Where ( / ) represents a vector of the Eigen values and equals the squares of

The Equations of (5.3 and 5.4) can be used to derive the frequency equation of any system and it's given by [7]:

[K ] - wi2 × [M ] = 0

……......................…. (5.8)

Expanding the frequency equation will give an algebraic equation of an (nth) degree that is known as the characteristic equation of the system [94]. The (n) roots
2 2 of this equation of ( w12, w 2 , w32 , …. , wn ) represents the square of the circular

frequencies of (n) modes of vibration (or Eigen values) which are possible in the system, while the corresponding (n) eigenvectors of this Eigen problem represent the (n) mode shapes of vibration for the system. Eq. (5.8) above is solved using a program coded by MATLAB.

5.4 Eigen value-Eigenvector Evaluation
The required natural frequencies of vibration and the corresponding mode shapes can be determined based on one of the following methods [23]: 1. Vector iteration methods. 2. Transformation methods. 3. Polynomial iteration methods.

5.4.1 Vector Iteration Methods
Various vector iteration methods are based on properties of the Rayleigh quotient. For the generalized Eigen value problem of Eq. (5.6), the Rayleigh quotient Q(v) is given by [23]:

VT × K V Q(v) = T V × MV
80

……......................…. (5.8)

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Where (v) is an arbitrary vector which defines a mode shape. A fundamental property of the Rayleigh quotient is that, it lies between the smallest and the largest Eigen values, that is:

/1 : Q(v) : /n
based on this property [23].

……......................…. (5.9)

Power iteration, inverse iteration and the subspace iteration methods are

5.4.2 Transformation Methods
The basic concept of this method is to transform the matrices to a simpler form and then to determine the Eigen values and the eigenvectors. The major methods in this category are the generalized Jacobi method and QR method. In QR method, the matrices are first reduced to tri-diagonal form using the Householder matrices. The generalized Jacobi uses the transformation to simultaneously diagonal the stiffness and mass matrices. This method needs the full matrix locations and is quite efficient for calculating all Eigen values and eigenvectors for small problems.
81

An advantage of the polynomial iteration method is that, an Eigen value is determined independently and cumulative errors do not occur. However, there is no guarantee that a value of (4) has been calculated without recourse to other methods, e.g. Strum Sequence checks. The Strum Sequence property of the characteristic polynomial is as follows:

Considering the effectiveness of the solution procedures, these methods are always the most efficient, but the solution technique used should be selected according to the specific problem to be solved.

5.5 Restriction of Supports
For simply supported and single span bridge decks where a partial restraint condition is encountered, the proposed Panel Element method (PEM) provides good technique as the Finite Element idealization method (FEM). None of all of the reliable other methods is applicable to idealize the condition of a partially restraint bridge deck at the supports as the real behavior.

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Generally, the supporting condition of any bridge deck might conform to one of two cases: 1. Partially restrained; that means, the end diaphragm is partially restrained in its out-of-plane direction, that is, only degrees of freedom (d.o.f) at the basic nodes are restricted as shown in Fig (5.1a). 2. Fully restrained; that means, the end diaphragm is fully restrained in its out-of-plane direction, that is, all longitudinal degrees-of-freedom (d.o.f) are restricted as shown in Fig (5.1b). The effects of the restrained condition on the natural frequencies and the corresponding mode shapes in free vibration response is studied in the illustrative case studies in the following sections.

Free d.o.f

Restricted d.o.f

(a) Partially Restrained Support

(b) Fully Restrained Support

Figure (5.1): Types of Supports Restrained Conditions. 83

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5.6 Description Of Case Studies
The case studies which are presented in this work, include two types of curved bridges, and two types of cellular deck bridge, and two types of the cross-sections of the box-bridge. The types of studied curved bridges are as follows: The first type of bridge is a curved bridge with (20m) span lengths, and acute angle of (206 degree) and radius of curvature (57.3m), as shown in Figs. (3.4c and 3.6c), while the second type of curved bridge is (30m) span length, with obtuse angle of (306 degree) and radius of curvature (57.3m), as shown in Figs. (3.5c and 3.7c). The types of studied cellular decks are as follows: the first type is a single cell as shown in Figs. (3.4d and 3.6d), while the second is a double cell curved deck as shown in Figs. (3.5d and 3.7d). Types of studied cross-sections of the box-bridge are as follows: the first type is a rectangular cross-section, as shown in Figs. (3.4a and 3.6a), while the second is a trapezoidal cross-section curved deck, as shown in Figs. (3.5a and 3.7a). Typical layout and cross-section dimensions for each type are shown in Figs. (3.4, 3.5, 3.6 and 3.7), and the material properties are listed in Tables (A-1 and A-2) in Appendix (A). These decks are the major decks of an existing bridge at Baghdad city (Ur-Qaherah bridge). For the purpose of the present analysis, it is assumed that each bridge is of a single (simply supported) span of length and central angle shown in Figs. (3.4c and 3.5c). It is worth mentioning that each deck is curved in the horizontal direction by a central angle ("=206 and "=306), and in the vertical direction by the profile described in Fig(3.4b and 3.5b). For comparison purposes, the decks of all cases, were modeled by adequate panel elements, that is, (12) degrees of freedom (d.o.f), which were sufficient to idealize the complete behavior of the bridge deck for its free vibration characteristics. Meanwhile, the same deck (single cell) and double cell (multi-cell), when using the Finite Element Method (FEM) procedure through the software (ANSYS), more elements, and more degree of freedom (d.o.f) were needed to predict the global free vibration characteristics of the bridge.

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5.7 Validation Case Studies
Several case studies concerning different bridge deck configurations were carried out and are presented in this chapter. The objective of these studies is to highlight the accuracy and reliability of the proposed idealization procedure in evaluating the behavior of cellular decks having different properties. Moreover, some important parameters and aspects including element type, mesh size, number of cells, cross-section of the deck, span length, web to slab thicknesses (tw/ts) ratio, diaphragms numbers (panels numbers) and live loading, are studied to give an idea about the effects of variation of these factors on the natural frequencies of the bridge. In the present work, the element type which is shell element, mesh size was varied for the curved deck bridge (coarse and fine), the number of cells was varied (1-cell and 2-cell) for both cases of curved deck bridge, the cross-section of the curved deck bridge was varied (rectangular and trapezoidal) for both cases of cellular type, the span length was considered varied (20m and 30m), the aspect ratio of (tw/ts) was varied from (0.5, 1, 1.5 and 2), for both cases of curved bridge deck (single and double cell), the diaphragms number (number of panels) was varied from (2, 4, 6 and 10) for both cases of curved bridge decks (single and double cell), finally the live load included three cases of loading that are described in section (5.8.7). Such variation resulted in a noticeable change in the natural frequencies. Appendix (B) shows sketches and tables for all numerical case studies. The tables show the total number of degrees of freedom and elements for both the Panel Element Method (PEM), the Finite Element Method (FEM) and the classical 3D beam element. All bridge decks, which are studied in the following numerical case studies, are assumed to be of reinforced concrete, which is assumed to be a linearly elastic material. The required material properties in the analysis are shown in tables in Appendix A (A1 & A2). The natural frequencies and the corresponding mode shapes, as a solution for the equation of motion given in Eq. (5.3) are evaluated using the inverse iteration scheme. For such purpose, the mass matrix of the bridge was formulated on the basis of its own weight of the deck for the first cases. The effect of live load is considered in the last case study by assuming equivalent masses to be lumped at the position where the live loads are located.
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5.8 Parametric Studies
To provide a better idea about the behavior of curved box girder bridge, some parametric studies are carried out including four main factors of; number of cells, web to flange thickness ratios and number of diaphragms along the span, in addition to the magnitude of live load acting on the deck, as follows:

5.8.1 Effect Of Number of Cells
First, two types of cell bridges (single and double) are considered for verification purposes. A cantilever deck of (20 m) span length, (2.3 m) depth and (3 m) width of each cell and element thicknesses of (0.3 m) is studied for its natural frequencies. The resulting first five natural frequencies are shown in Tables (5-5 and 5-6). The proposed panel element (PE) method has proved to be appropriate method for predicting the natural frequencies of a single and a double deck bridges with a maximum differences in the fundamental mode for a double cell deck in the range of (2%) and in the fifth mode of about (8%) only for both types of cross-sections (rectangular and trapezoidal). The major conclusion drawn from these specific case studies is increasing the number of cells results in reduction in the values of natural frequencies. Such a behavior can be related to the fact that an increase in the width of a bridge deck, results in an increase in the lateral stiffness and the total mass, that is, no increase in the dynamic stiffness, therefore, resulting in reducing the natural frequencies in this case.

5.8.2 Effects Of Web To Flange Thicknesses Ratio
Second, a single cell deck is considered with a depth of (2.3 m) and a width of (3.0 m), then a double cell deck is considered with a depth of (2.3 m) but a width of 6.0 m (3.0 m for each cell). The bridges under consideration are of (20 m) single span with partially and fully restrained supports, the radius of curvature of the bridge span is (57.3 m). The thickness ratios varied from (0.5 to 2.0). The resulting first
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five natural frequencies are listed in Tables (5-11, 5-12, 5-13 and 5-14), and the plots are presented in Figs. (5.2, 5.3, 5.4 and 5.5) for both a single and a double cell decks, and for partially and fully restrained supports, respectively. The study showed that as the web thickness to slab thickness increases, natural frequencies of the bridge increase too. This can be attributed to the increase in the bridge stiffness, especially, in lateral-torsional modes.

5.8.3 Effects Of Number Of Diaphragms
In this case, the number of diaphragms represents the number of panels where each panel represents a segment between two diaphragms. The numbers of panels are changed from (2 to 10) to demonstrate the effect of variation in number of diaphragms for a constant span length on the free vibration characteristics. The natural frequencies which are obtained from the free vibration response by the panel element (PE) idealization method are listed in Tables (5-15, 5-16, 5-17 and 5-18) and the plots are shown in Figs. (5.6, 5.7, 5.8 and 5.9) for both partially and fully restrained supports, respectively. The parametric study demonstrates that the proposed panel element (PE) approach predicts the free vibration characteristics more accurately where the number of diaphragms (number of panels) increases.

5.8.4 Element Type Effect
Two cases of bridges, namely, a cantilever and a simply supported single cell deck of (20 m and 30 m) span length, (0.3 m) web and flange thicknesses, (2.3 m) depth and (3 m) width are analyzed using the proposed (PE) approach and compared with the shell element (FE) and 3D beam element given in ANSYS. The resulting natural frequencies are shown in Tables (5-1 and 5-2). The proposed (PEM) idealization procedure proved to be acceptable in predicting the free vibration response of both types of bridge decks (the cantilever and the simply supported) for both spans as compared to the finite element (FE) approach using shell elements with differences of not more (7%) for the first two
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modes of vibration. However, both, the Panel Element Method (PEM) and the Finite Element Method (FEM) showed large differences in the natural frequencies in comparison to the approximate solution based on the 3D beam element. This difference is dependent on the element type with the selecting of elements number; in which, an increase in the number of elements, mean an increase in the degrees of freedom (d.o.f), and these caused an increase in the vibration of the bridge; therefore, resulting in increasing the natural frequencies values.

5.8.5 Mesh Size Effect
Other numerical case studies for verification purposes are considered. A cantilever and simply supported single cell deck of (20m and 30m) span length, (2.3m) depth and (3m) width, and a slab and web thickness of (0.3 m) are considered. These two bridge cases were analyzed for their natural frequencies using the new proposed panel element method (PEM) together with the finite element method (FEM). In each case, the number of panel segments and the mesh size were progressively refined till convergence of solution is reached. Results are presented in Tables (5.3 and 5.4). Convergence of the panel element method (PEM) seems to be much faster as compared to the finite element (FE) approach where only (3%) difference is occurred in increasing the number of segments by (25%).

5.8.6 The Cross-Section Effect
The last verification examples are two curved bridge decks, one of (20 m) and the other of (30 m) span, both are analyzed for two cases, a single and a double cell decks. For each numerical case study, the cross-section is first assumed to be rectangular and then assumed to be trapezoidal. Two boundary conditions are assumed, partially restrained ends and fully restrained ends (at supports). Results are presented in Tables (5-7, 5-8, 5-9 and 5-10). In case of partially restrained supports, the fundamental natural frequency as predicted by the developed panel element method (PEM) is found to be of
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less than (1%) difference from that obtained by the finite element (FE) approach for the case of single cell curved bridges of rectangular and trapezoidal cross-sections as compared to the difference of (7%) at higher modes (in case of partially restrained ended bridge). Differences in natural frequencies were found to increase slightly (but less than 9%) in cases of double cell curved bridges for the cases of rectangular and trapezoidal cross-sections. When the bridges become fully restrained at their ends, the encountered differences between the panel element method (PEM) and the finite element (FE) approach are reduced to less than (3%). In generally, this change is dependent on the cross-section type with the boundary condition of the end supports (partially or fully restrained), such a behavior can be related to the fact that an increase in the un-uniformly of cross-section, results in an increase in the vibration of the body, and leads to change in the stiffness and the total mass, that is, caused by increase in the dynamic stiffness, therefore, resulting in increasing the natural frequencies values. It was also found that, the assumption of fully restrained support result in over estimation of the dynamic stiffness of bridge decks and hence, the resulting natural frequencies are higher than the real values.

5.8.7 Effect of Live Load
Finally, to explain the effect of live load, simple load cases are considered according to the Iraq's Specifications for Bridge Loading [72], and as follows: 1. Lane loading, where loads are distributed uniformly over the deck and knife edge load is considered at mid-span to give the maximum response. This load condition is designated the fast load case (load case I). 2. Military Loading, two classes of this loading are studied as follows: i. Class 100 (Tracked), one tracked at mid-span. This load condition is designated the second load case (load case II). ii. Class 100 (Wheeled), one wheeled at mid-span. This load condition is designated the third load case (load case III).
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The resulting natural frequencies are listed in Tables (5-19, 5-20, 5-21 and 5-22). The proposed Panel Element Method (PEM) showed good agreement with the result obtained by the finite element (FE) approach. All the above mentioned numerical case studies resulted in the main conclusion that the developed panel element (PE) approach can be regarded a versatile procedure for evaluation of the free vibration characteristics of curved bridge decks having single and double cells in cross-sections of different span lengths and of partial or full restrained conditions at supports. Though the finite element (FE) approach is a well known reliable approach the proposed panel element (PE) approach is promising according to the economic solution where more than (90%) reduction in the number of degrees of freedom (d.o.f) is gained when using the panel element (PE) procedure. This important saving in the number of equations as compared to the minor sacrifice in accuracy of the first few modes of vibration can be considered a major outcome of the developed technique. Some mode shapes of single and double cell bridge decks of a cantilever and fixed ended spans are shown in Figs. (5.10, 5.11, 5.12 and 5.13).

characterized by time dependent amplitudes and frequencies. From the past historical records of earthquake occurrence, it has been seen that earthquake is one of the most feared natural disasters which has caused incalculable destruction of properties and injury and loss of lives to the population. Earthquakes occur due to the instability of the earth crust and the sudden release of accumulated stress deep inside the crust. The sudden release of energy during an earthquake may lead to ground shaking, surface faulting, and ground failures. Stresses are generated in structures due to the ground shaking and if a structure is incapable of resisting these additional stresses, it will suffer damage. The current philosophy behind earthquake resistant design of common structures is to ensure that: [122] i) Hazards to life is minimized. ii) Design ground motions have low probability of being exceeded during normal lifetime of bridge. iii) Function of essential bridges is maintained. iv) There are no damages (or only slight but reparable nonstructural damage) because of design earthquakes. Bridges may suffer damage but have low probability of collapse due to earthquake motion. v) Collapse is prevented during more severe earthquakes, which is achieved by ensuring ductile, rather than brittle behavior of the structural response.
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6.2 Scope and Objective
This chapter includes the dynamic analysis of curved cellular bridges subjected to earthquake base excitation, which is characterized by two components, each of which is transverse to the longitudinal axis, but, one is assumed to be in the vertical direction and the other in the horizontal direction. The 20 Feb. 1990 modified smooth pseudo-acceleration design spectrum of AL-Hindya Earthquake, which is shown in Fig. (4.19), characterizes each base excitation component [5]. Due to the lack of acceleration records in the vertical direction, the same response spectrum that is used in the analysis of bridge decks is acted upon by horizontal base excitation, the same spectrum is assumed for the vertical earthquake analysis, or using El-Centro components (scaled down). It is also assumed that, the motion of all supports of the bridge have in phase excitation, that is, all supports are acted upon by the same base excitations simultaneously. Since all structures considered in this study have two axes of symmetry, any base excitation component produces no torsional response, that is, only lateral response is produced. The objective of the present research is to analyze the longitudinal and transverse earthquake motion of the bridge and to determine the design forces and moments at the supports bases by the Finite Element Method (FEM) and the Panel Element Method (PEM). Then a comparative study of the design forces and moments found from these two methods has been made. It is expected that the findings of this study will lead to a better understanding of the behavior of bridges under seismic loading. For simplicity of the analysis, linear material behavior is assumed in this study.

6.3 Dynamic Analysis
In the dynamic analysis, the main techniques currently used are: 1- Response Spectrum Techniques This method is based on the mode superposition approach. The general procedure is to compute the response of each of the structure's individual modes and then to combine these responses to obtain the overall response.
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Only few modes can be included in computing any particular response of the system. The specific modes, which must be considered, depend upon the properties of the structures and the particular quantity, which is being computed, that is, the modal mass participation factors in each mode. 2- Direct Integration Techniques: Direct integration of the equations of motion, is based on step by step procedure. This technique is general for any earthquake motion action on structure, for which a moment and force diagrams, at each of the series of prescribed intervals throughout the applied motion, can be found for both linear elastic and nonlinear elastic material behavior. This method is the more nearly complete dynamic analysis technique so far devised, and is unfortunately correspondingly expensive to carry out. 3- Normal Mode Technique: The normal mode technique is a more limited approach than the direct integration, as it depends on an artificial combining the forces and displacements associated with a chosen number of them using modal superposition.

6.4 Equation of Motion
In a typical dynamic problem, the motion of a system excited by an external dynamic load, and the complete set of forces acting on the system in additional to the external forces are; inertia, damping and elastic forces which resist the motion and are proportional to the accelerations, velocities and displacements of the system, respectively. Thus, the equation of dynamic equilibrium of all forces acting on a multi-degree of freedom system at any time (t), can be written as follows:

. [M] . {'} + [C] . {,} + [K] . {U} = - [M] . {R} . 'g

………. (6.1)

Where {R} denotes an earthquake influence vector consisting of ones and zeros, where ones correspond to the degree of freedom in the direction of the base excitation component, and zeros elsewhere. Equation (6.1) above, can be transformed to the normal coordinates, to give the vibration mode of the structure, that is:
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Earthquake Response Analysis …..................…. (6.2)

l 2 && (t ) &&n + 2zw n × U & n + wn U × Un = nr × U gr Mn
Where:

M n = FT n × M × Fn l nr = F T n × M × Rr

, is the generalized mass at mode (n). , is the modal earthquake excitation factor in the r-direction (r = X, Y).

If the response spectrum of the ground motion is available, the maximum response of the system at each node can be obtained from it, depending upon the natural time period and the damping ratio of the structure. The response could be the spectral acceleration, velocity or displacement, for which:

The elastic forces (Fs) associated with the relative displacements can be obtained directly by pre-multiplying the relative displacements by the stiffness matrix, such that:

Fs (t) = K . U(t) = K . 9 . Y(t)
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………………………. (6.8)

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Earthquake Response Analysis

It is more convenient to express these forces in terms of the equivalent inertia forces developed in the undamped vibration [34] such that:

{F }= w × [M ]× {U }
Snr 2 n nr

= [M ]× {F n }×

l nr × Sa r (z , T) Mn

…....................................…. (6.9)

6.5 Structural Response
It should be pointed out that, in practice, the superposition of the mode responses is usually done in one of three ways [43]. The most conservative approach that yields an upper limit is to add numerically the response of the modes. This approach yields reasonable results for cases where the contribution of the fundamental mode predominates. For many problems this is usually true. A less conservative approach is to take the sum of the fundamental mode response plus the square root of the sum of the squares of the higher modes. This will yield more reasonable results for cases where the contributions of the higher modes are appreciable. A third approach is to obtain a total maximum response by taking the root mean square, that is, the square root of the sum of the squares of the maximum responses. The third approach is considered in the present study thus, the maximum forces are approximated by:
2 2 FSmax = ( FS1 ) 2 max + ( FS 2 ) max + LLLL + ( FSn ) max

Where [K] denotes the stiffness matrix of the overall structure after applying boundary conditions and {e}, denotes the displacement vector produced by the static analysis of bridge deck subjected to the maximum force vector {FSmax} which are obtained from the response spectrum analysis. Back substitution is applied to evaluate the reactions at supports.
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6.6 Numerical Case Studies
The same examples of bridge decks that were analyzed for their free vibration characteristics are next analyzed for their earthquake response when acted upon by a horizontal base excitation (orthogonal to the longitudinal tangential axis of the bridge at its mid-span) and a vertical base excitation. All these bridges are analyzed using both the proposed Panel Element (PE) approach and the results are compared to those obtained by the Finite Element (FE) procedure using the ready software ANSYS. The resulting moments and shear forces of each case study is given in two ways and as follows: 1- Absolute response that is, the maximum moment and maximum shear force. 2- Normalized response, that is, the resulting moment and shear forces are normalized to: i. Total mass of the bridge (m*). ii. The product of (m*) times the span length of the bridge (L).

6.7 Parametric Studies
Some parametric studies are carried out including four main factors of; number of cells, web to flange thickness ratios, number of diaphragms and live load effects, to provide a better idea about the behavior of curved box girder bridges, as follows:

6.7.1 Effects of Number of Cells Variation
The same two bridge decks of section 5.8.1 (curved bridge decks one of a single cell and the second of a double cell) with the same dimensions, shapes and central angle and radius of curvature are considered here. The absolute and normalized moments, shear forces and deflections of a cantilever deck are given in Tables (6-1 and 6-2), and the plots are presented in Figs. (6.1 and 6.2) for base excitation in (X and Y-directions), respectively.

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Analysis showed that a maximum difference of less than (12%) in deflection, base shears and bending moments is encountered between the proposed Panel Element Method (PEM) of analysis and the finite element (FE) procedure of ANSYS software irrespective of the direction of earthquake base excitation.

6.7.2 Effects Of Web To Flange Thicknesses Ratio
The same bridge decks that are presented in section (5.8.2) are studied here for their earthquake response characteristics. To demonstrate the range of applicability of the proposed idealization procedure of the Panel Element Method (PEM) to different ranges of (web thickness: Slab thickness) ratio. The single cell bridge deck fully restrained at supports which is discussed in chapter five is used here for earthquake response in two directions (X and Y- directions) transverse to the longitudinal axis of the deck (Z-axis). The results are presented in Tables (6-3 and 6-4), and the plots are shown in Figs. (6.3 and 6.4) for the cases of a single cell and double cell bridge decks. Results reveal that the proposed idealization procedure of the Panel Element Method (PEM) works well in evaluating the response of bridge decks subjected to earthquake base excitation as compared with the Finite Element Method (FEM) with errors not more than (10%) in the deflection and no more than (17%) in moments and shear forces when the (tw/ts) thickness ratio reaches (2).

6.7.3 Effects Of Number Of Diaphragms
Next is a parametric study on the effect of number of diaphragms along the constant span length on the earthquake response behavior of curved bridges. The same curved bridges considered in the free vibration analysis are considered here to demonstrate the range of applicability of the proposed panel element (PE) idealization scheme for two cases of earthquake base excitation, namely; in a horizontal direction normal to mid-span tangent and in the vertical direction.

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Maximum moments and deflections at mid-span and maximum shear forces at supports as a response of the bridge deck when acted upon by base excitation are shown in Tables (6-5, 6-6, 6-7 and 6-8), and the plots are presented in Figs. (6.5, 6.6, 6.7 and 6.8) for both partially and fully restrained support conditions for earthquake base excitation in X and Y-directions, respectively. It can be seen clearly from the results that variation of number of panels results in significant change in the value of the deflection for both directions and boundary condition types, with errors not more than (12%) in the deflection and no more than (18%) in moments and shear forces when the number of diaphragms reaches (10). It is also concluded that the proposed idealization procedure of the Panel Element Method (PEM) is valid for all the range of numbers of diaphragms considered in the study.

6.7.4 Effect of Live Load
The same load cases, which were considered in the free vibration analysis, are adopted here. The uniformly distributed lane load is considered as an additional mass added to the mass density of the structures. The other types of live loads are considered as lumped masses added to the corresponding degrees of freedom in lateral directions (X and Y-directions). All the results, which represent the dynamic response of the bridge deck subjected to earthquake base excitation in X and Y-directions and for both partially and fully restrained boundary conditions are given in Tables (6-9, 6-10, 6-11 and 6-12). It can be seen from the above mentioned tables a good agreement with response predicted by the finite element is demonstrated out of these numerical case studies, with errors not more than (10%) in the deflection and no more than (16%) in moments and shear forces. Also, the tables give another evidence of the validity of the proposed idealization procedure of the Panel Element Method (PEM).
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CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions
In order to assess the efficiency and accuracy of the proposed idealization procedure designated the Panel Element Method (PEM) for free vibration and earthquake response analysis of curved cellular box-girder deck bridge structures, a number of examples of case studies are analyzed. Different configurations of curved cellular bridge decks are considered to verify the proposed Panel Element Method (PEM) against the Finite Element Method (FEM) for both free and forced vibrations. According to the case studies considered in the present research, the major conclusions are drawn, as follows: 1. The Panel Element Method (PEM) of idealizing curved cellular type bridge decks is verified for the dynamic analysis of free vibration and earthquake response of almost all-practical deck configurations, which are considered. The proposed idealization technique of the Panel Element Method (PEM) can predict accurately (as compared to the Finite Element (FE) procedure) the free vibration characteristics (natural frequencies and mode shapes) of single and double cell curved box-girder bridge decks, and for rectangular and trapezoidal cross-sections. The difference in the fundamental modes of vibration (natural frequency) is less than (7%) between the proposed panel element (PE) and the standard finite element (FE).
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2. At the same time, a varied reduction in the number of elements and hence, the degrees of freedom (d.o.f) is gained when using the proposed idealization procedure of panel element (PEM) to model the behavior of the bridge under consideration as compared to the traditional finite element (FE) procedure. 3. Moreover, since the number of degrees of freedom as needed by the proposed element is limited, the number of equations and iterations are largely reduced and hence less error is encountered as shown in the tables in Appendix (B). 4. Elimination of all local and intermediate degrees of freedom (d.o.f) (as the degrees of freedom (d.o.f) are defined only at locations of panel ends and diaphragms) will result in a versatile solution by eliminating all local vibration modes (fictitious or unreal modes), thus, the free vibration response can be evaluated accurately and through less computational efforts. 5. A parametric study on the variation of the number of diaphragms (number of panels) showed the natural frequencies are more sensitive to the change in number of diaphragms, which is, convergence of solution is reached faster than in the finite element (FE) approach. 6. The Panel Element Method (PEM) has proved to be valid in estimating the earthquake response for both cases of single and double cell bridge decks. 7. For all the ranges of the aspect ratios; the results obtained by the Panel Element Method (PEM) are acceptable, with an error of less than (12%) in deflection and less than (18%) in moments and shear forces for the cases of very large aspect ratios. It can be seen that the proposed Panel Element Method (PEM) predicts good estimates of cases of small aspect ratios. The normalized value shows good compatibility in response especially in the values of moments when normalized to the product of total mass and the span of the decks.

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Conclusions & Recommendations

8. Variation of the number of diaphragms for a constant span length of a bridge deck results in almost no change in the deflection. The errors encountered in estimating the deflection of bridge decks are inversely proportional to the number of panels. The number of diaphragms has proved to be of insignificant influence on the moment and shear force responses of curved bridge decks when acted upon by earthquake base excitations.

7.2 Recommendations For Future Works
The Panel Element Method (PEM) proved to be applicable to analyze the dynamic behavior (free and forced vibrations) of almost all-practical bridge deck configurations. Therefore, it provides an efficient alternative to the Finite Element Method (FEM) for providing acceptable response estimates for complex structure configurations such as support conditions, which cannot be represented by some other existing idealization procedures. Therefore, it is recommended to investigate the efficiency of the proposed Panel Element Method (PEM) idealization scheme in the following cases as future works: 1. A parametric study on the effect of the number of the interior webs on the deflection, stresses developed within the bridge. 2. Cases of continuous curved box-girder bridges, and torsional effect due to eccentricity of the loading. 3. Investigate the soil-structure interaction problem during strong earthquake motions. 4. It is recommended also to study the effect of geometric and material nonlinearities on the earthquake response analysis of curved cellular box-girder bridges. 5. Finally, it is recommended that more different configurations of curved cellular box-girder bridge decks are to be studied, such as, those with prestressed concrete.